1
AB = 8 cm, BC = 15 cm, AC = 17 cm, right-angled at B. sin A is —
💡 Hint:
Opposite over hypotenuse.
Explanation:
BC/AC = 15/17.
Class 10 · Maths · Chapter 8 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Introduction to Trigonometry
Introduction to Trigonometry
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1. Read — sin cos tan · reciprocals · 0° to 90° · sin²+cos²=1 · sec²−tan²=1, diagram, worked example, board tip
2. Check — each lesson has its own questions; the number follows the lesson
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4. The memory figure shows the opposite, the adjacent and the hypotenuse in a right triangle and writes sin A = opposite/hypotenuse.
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- 1 sin, cos and tan
- 2 Reciprocals, and the rest from one ratio
- 3 Values at 0°, 30°, 45°, 60°, 90°
- 4 Adding and subtracting the values
- 5 sin²A + cos²A = 1
- 6 The identities with sec and cosec
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
sin, cos and tan
sin, cos और tan · NCERT 8.2 · Exercise 8.1 · opposite and hypotenuse
Three sides, one angleNotes
The triangle is right-angled. Angle A is acute. sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent.
The hypotenuse is the longest side and it faces the right angle. These ratios are not formed for the right-angled vertex. Exercise 8.1 first asks you to read the three ratios from the sides. Also tan = sin/cos, when cos is not zero.
Mark the right angle firstActivity
If the sides are 8 cm, 15 cm and 17 cm and the right angle is at B, the side opposite A is BC = 15 cm, the adjacent side is AB = 8 cm and the hypotenuse is AC = 17 cm. 8² + 15² = 64 + 225 = 289 = 17². Do not write a ratio before this check.
Worked exampleExample
Question: The right angle is at B. AB = 8 cm, BC = 15 cm, AC = 17 cm. Find sin A, cos A and tan A.
Formula: sin A = BC/AC, cos A = AB/AC, tan A = BC/AB.
Substitution: sin A = 15/17, cos A = 8/17, tan A = 15/8.
Answer: sin A = 15/17, cos A = 8/17, tan A = 15/8. A ratio has no unit left. Check: (8/17)² + (15/17)² = (64+225)/289 = 1.
10-second revision
- sin = opposite/hypotenuse
- cos = adjacent/hypotenuse
- tan = opposite/adjacent
Board tip · BSEBBoard tip
In a BSEB answer write the side in cm and the ratio as a fraction. Do not give both the same unit.
Board tip · CBSEBoard tip
On CBSE do not turn the right-angled letter into angle A. Keep the letter in the figure.
Check your understandingall correct = mastery ★
2
In the same triangle cos A = 8/17.
Explanation:
True — adjacent/hypotenuse = AB/AC.
3
In the same triangle tan A = ______.
💡 Hint:
Opposite over adjacent.
Explanation:
15/8.
4
In the same triangle write sin C and cos C.
Model answer (board format)
Opposite C is AB = 8 cm and adjacent to C is BC = 15 cm. sin C = 8/17, cos C = 15/17.
Check your answer:
Reciprocals, and the rest from one ratio
व्युत्क्रम और एक अनुपात से बाकी · Exercise 8.1 · cosec, sec, cot · sin ≤ 1
A whole triangle from one fractionNotes
cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A. For an acute angle all of them are positive.
If sin A = 3/5, the opposite side is 3, the hypotenuse is 5 and the adjacent side is √(25−9) = 4. Then cos = 4/5 and tan = 3/4. sin or cos is never greater than 1. sin θ = 4/3 is impossible. sec = 13/5 is possible, because sec is at least 1. cos is not the name of cosecant. The true-or-false part of exercise 8.1 is these cautions.
Keep the hypotenuse as the longest sideActivity
If cot A = 8/15, the adjacent side is 8 and the opposite side is 15. The hypotenuse is √(64+225) = √289 = 17. sin A = 15/17 and sec A = 17/8. If someone says the hypotenuse is 8, the triangle cannot exist.
Worked exampleExample
Question: sin A = 3/5 and A is acute. Find cos A and tan A.
Formula: cos²A = 1 − sin²A, and tan A = sin A / cos A.
Substitution: cos²A = 1 − 9/25 = 16/25. cos A = 4/5, because cos is positive for an acute angle. tan A = (3/5)/(4/5) = 3/4.
Answer: cos A = 4/5, tan A = 3/4. Check: opposite 3, adjacent 4, hypotenuse 5.
10-second revision
- cosec = 1/sin, sec = 1/cos, cot = 1/tan
- sin and cos are at most 1
- For an acute angle the square root is positive
Board tip · BSEBBoard tip
In a BSEB answer show the 3-4-5 triangle on one line, then the ratio.
Board tip · CBSEBoard tip
On CBSE it is not enough to call sin = 4/3 a large value. Write that it is impossible.
Check your understandingall correct = mastery ★
1
If sin A = 3/5, then cos A is —
💡 Hint:
Adjacent 4, hypotenuse 5.
Explanation:
4/5.
2
sin θ can be 4/3 for some angle.
Explanation:
False — sin is never greater than 1.
3
If sin A = 3/5, then cosec A = ______.
💡 Hint:
The reciprocal.
Explanation:
5/3.
4
If cot A = 8/15, the hypotenuse is —
💡 Hint:
8² + 15².
Explanation:
√289 = 17.
5
cos A is the short name of cosecant.
Explanation:
False — cos is cosine. The name of cosecant is cosec.
6
sec A = 13/5 and A is acute. Find sin A.
Model answer (board format)
Hypotenuse 13, adjacent 5. Opposite √(169−25) = √144 = 12. sin A = 12/13.
Check your answer:
Values at 0°, 30°, 45°, 60°, 90°
0°, 30°, 45°, 60°, 90° के मान · NCERT 8.3 · the table for exercise 8.2
Five angles, three rowsNotes
The sin row is 0, 1/2, 1/√2, √3/2, 1. The cos row is the reverse. tan = sin/cos.
In the isosceles right triangle at 45°, if both legs are a then the hypotenuse is a√2, so sin 45° = cos 45° = 1/√2 and tan 45° = 1. The values 30° and 60° come from cutting an equilateral triangle in half. tan 90° and sec 90° do not exist. cot 0° and cosec 0° do not exist. Exercise 8.2 reads this table.
Write sin, then read cos backwardsActivity
Write sin under the angles 0, 30, 45, 60, 90. The same five values read from right to left become the cos row. tan 30° = (1/2)/(√3/2) = 1/√3. tan 60° = (√3/2)/(1/2) = √3. In the table both sin 30° and cos 60° are 1/2. That is not an addition rule.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | not defined |
Worked exampleExample
Question: Write sin 30° and tan 45°. Also cos 60°.
Formula: From the table — sin 30° = 1/2, tan 45° = 1, cos 60° = 1/2.
Substitution: No side is given. The values come straight from the table.
Answer: 1/2, 1 and 1/2. sin 30° = cos 60°.
10-second revision
- sin 30° = 1/2
- tan 45° = 1
- tan 90° is not defined
Board tip · BSEBBoard tip
In a BSEB answer you may also write 1/√2 as √2/2, but keep one form through the answer.
Board tip · CBSEBoard tip
On CBSE do not write tan 90° as 0 or as a number called infinity. Write that it is not defined.
Check your understandingall correct = mastery ★
1
sin 30° equals —
💡 Hint:
The second angle in the table.
Explanation:
1/2.
2
tan 45° = 1.
Explanation:
True — the opposite and adjacent sides are equal.
3
cos 60° = ______.
💡 Hint:
Equal to sin 30°.
Explanation:
1/2.
4
tan 90° —
💡 Hint:
cos 90° = 0.
Explanation:
tan = sin/cos, and the denominator is zero.
5
Why are sin 45° and cos 45° equal? Write one reason.
Model answer (board format)
In a 45°-45°-90° triangle the opposite and the adjacent sides are both a and the hypotenuse is a√2. So both ratios are 1/√2.
Check your answer:
Adding and subtracting the values
मानों को जोड़ना और घटाना · Exercise 8.2 · sin(A+B) ≠ sin A + sin B
A calculation from the table, not a new ruleNotes
From 0° to 90°, sin increases and cos decreases. sin(A + B) is not sin A + sin B.
sin 60° cos 30° + sin 30° cos 60° = 1, which is sin 90°. But sin 60° + sin 30° = √3/2 + 1/2, which is not 1. sin θ = cos θ only at 45°, not at every angle. cot 0° does not exist. The true-or-false part of exercise 8.2 is this.
Turn each value into a fraction, then addActivity
2 tan² 45° + cos² 30° − sin² 60°. tan 45° = 1, so the first term is 2. cos 30° = √3/2, and its square is 3/4. sin 60° = √3/2, and its square is 3/4. 2 + 3/4 − 3/4 = 2. Do not put the square on the angle.
Worked exampleExample
Question: Evaluate sin 60° cos 30° + sin 30° cos 60°.
Formula: sin 60° = cos 30° = √3/2, sin 30° = cos 60° = 1/2.
Substitution: (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1.
Answer: 1. This equals sin 90°. sin 60° + sin 30° is not this value.
10-second revision
- From 0° to 90°, sin rises and cos falls
- Do not split sin(A+B) into a sum
- sin equals cos only at 45°
Board tip · BSEBBoard tip
In a BSEB answer write each standard value on its own line, then add in one fraction.
Board tip · CBSEBoard tip
On CBSE write the sentence “increases” together with the range 0° to 90°.
Check your understandingall correct = mastery ★
1
sin 60° cos 30° + sin 30° cos 60° equals —
💡 Hint:
3/4 + 1/4.
Explanation:
1.
2
sin(A + B) = sin A + sin B for every acute angle.
Explanation:
False — at 60° and 30° the left side is 1 and the right side is not.
3
2 tan² 45° + cos² 30° − sin² 60° = ______.
💡 Hint:
2 + 3/4 − 3/4.
Explanation:
2.
4
From 0° to 90°, does cos decrease or increase? Also compare sin 0° and sin 90°.
Model answer (board format)
cos 0° = 1 and cos 90° = 0, so cos decreases. sin 0° = 0 and sin 90° = 1, so sin increases.
Check your answer:
sin²A + cos²A = 1
sin²A + cos²A = 1 · NCERT 8.4 · Exercise 8.3 · the first identity
Divide Pythagoras by the square of the hypotenuseNotes
In a right triangle, opposite² + adjacent² = hypotenuse². Divide every term by hypotenuse². sin²A + cos²A = 1.
This is true for every angle from 0° to 90°, so it is an identity. An equation that is true for only one angle is not an identity. Exercise 8.3 uses it to write the other ratios in terms of cot or sec. The square sits on the ratio, not on the angle: sin²A means (sin A)².
Subtract the known square to get the otherActivity
If sin A = 5/13, then sin²A = 25/169. cos²A = 1 − 25/169 = 144/169. cos A = 12/13. tan A = 5/12. Hypotenuse 13, opposite 5, adjacent 12. The 5-12-13 check is 25 + 144 = 169.
Worked exampleExample
Question: sin A = 5/13 and A is acute. Find cos A.
Formula: cos²A = 1 − sin²A.
Substitution: cos²A = 1 − 25/169 = 144/169.
Answer: cos A = 12/13. Not the negative root, because A is acute.
10-second revision
- sin²A + cos²A = 1
- The square is on the ratio, not on the angle
- For an acute angle cos is positive
Board tip · BSEBBoard tip
In a BSEB answer write 1 − 25/169 as the single fraction 144/169. Do not jump to 12/13.
Board tip · CBSEBoard tip
On CBSE do not read sin²A as sin of A².
Check your understandingall correct = mastery ★
1
If sin A = 5/13, cos A is —
💡 Hint:
The root of 144/169.
Explanation:
12/13.
2
sin²A + cos²A = 1 is true only at 45°.
Explanation:
False — it is an identity from 0° to 90°.
3
If sin A = 5/13, then cos²A = ______.
💡 Hint:
1 − 25/169.
Explanation:
144/169.
4
sin²A means —
💡 Hint:
The square is on the ratio.
Explanation:
The square of (sin A).
5
cos A = 12/13 and sin A = 5/13 can occur together for an acute angle.
Explanation:
True — 25/169 + 144/169 = 1.
6
If cos A = 8/17, find sin A and tan A.
Model answer (board format)
sin²A = 1 − 64/289 = 225/289. sin A = 15/17. tan A = (15/17)/(8/17) = 15/8.
Check your answer:
The identities with sec and cosec
sec और cosec वाली पहचान · Exercise 8.3 · 1 + tan²A = sec²A
Divide the same Pythagoras by another sideNotes
Divide Pythagoras by the square of the adjacent side. 1 + tan²A = sec²A. Divide by the square of the opposite side and you get 1 + cot²A = cosec²A.
So sec²A − tan²A = 1. Then 9 sec²A − 9 tan²A = 9. This is the first multiple-choice item of exercise 8.3. At 90°, tan and sec do not exist, and at 0°, cot and cosec do not exist. (sec A + tan A)(1 − sin A) = cos A, because (1 − sin²A)/cos A = cos A.
The identity first, then the numberActivity
9 sec²A − 9 tan²A = 9(sec²A − tan²A) = 9 × 1 = 9. The options are 1, 9, 8 and 0. The correct one is 9. Take the 9 outside before you subtract inside.
Worked exampleExample
Question: Find the value of 9 sec²A − 9 tan²A.
Formula: sec²A − tan²A = 1.
Substitution: 9(sec²A − tan²A) = 9 × 1.
Answer: 9. This holds for every A where sec and tan exist.
10-second revision
- 1 + tan²A = sec²A
- 1 + cot²A = cosec²A
- 9 sec²A − 9 tan²A = 9
Board tip · BSEBBoard tip
In a BSEB answer show the 9 outside the bracket. Jumping to 9 is thin.
Board tip · CBSEBoard tip
On CBSE write the identity only in the angle range where it exists. Do not use sec² − tan² at tan 90°.
Check your understandingall correct = mastery ★
1
9 sec²A − 9 tan²A equals —
💡 Hint:
9 times (sec² − tan²).
Explanation:
9.
2
1 + cot²A = cosec²A.
Explanation:
True — where cot and cosec exist.
3
sec²A − tan²A = ______.
💡 Hint:
The identity.
Explanation:
1.
4
(sec A + tan A)(1 − sin A) equals —
💡 Hint:
(1 − sin²A)/cos A.
Explanation:
cos²A / cos A = cos A.
5
If tan A = 3/4, find sec A. A is acute.
Model answer (board format)
sec²A = 1 + tan²A = 1 + 9/16 = 25/16. sec A = 5/4. Check: adjacent 4, opposite 3, hypotenuse 5.
Check your answer:
❓ Full question bank — with answers and explanations — 65 questions
Pick a type. The 30 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ120 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA120 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/12Pick one option. A wrong try brings a hint.
1
sin A equals —
💡 Hint:
The definition.
Explanation:
Opposite/hypotenuse.
2
AB = 8, BC = 15, AC = 17, right angle at B. cos A is —
💡 Hint:
Adjacent/hypotenuse.
Explanation:
8/17.
3
If sin A = 3/5, cos A is —
💡 Hint:
3-4-5.
Explanation:
4/5.
4
sin 30° is —
💡 Hint:
The table.
Explanation:
1/2.
5
tan 45° is —
💡 Hint:
Opposite equals adjacent.
Explanation:
1.
6
sin 60° cos 30° + sin 30° cos 60° is —
💡 Hint:
3/4 + 1/4.
Explanation:
1.
7
If sin A = 5/13, cos A is —
💡 Hint:
144/169.
Explanation:
12/13.
8
9 sec²A − 9 tan²A is —
💡 Hint:
9 × 1.
Explanation:
9.
9
tan 90° —
💡 Hint:
cos 90° = 0.
Explanation:
Not defined.
10
If sec A = 13/5, sin A is —
💡 Hint:
Opposite 12.
Explanation:
12/13.
11
2 tan²45° + cos²30° − sin²60° is —
💡 Hint:
2 + 3/4 − 3/4.
Explanation:
2.
12
If tan A = 3/4, sec A is —
💡 Hint:
1 + 9/16 = 25/16.
Explanation:
5/4.
True or false
0/81
cos A = adjacent/hypotenuse.
Explanation:
True.
2
sin θ = 4/3 is possible.
Explanation:
False — sin is at most 1.
3
tan A is always less than 1.
Explanation:
False — tan 60° = √3 > 1.
4
cos 60° = 1/2.
Explanation:
True.
5
sin(A+B) = sin A + sin B.
Explanation:
False.
6
sin²A + cos²A = 1.
Explanation:
True.
7
cot 0° is defined.
Explanation:
False — sin 0° = 0, so cot does not exist.
8
1 + tan²A = sec²A.
Explanation:
True, where both exist.
Fill in the blanks
0/81
tan A = opposite / ______.
💡 Hint:
Adjacent.
Explanation:
The side next to the angle.
2
If sin A = 3/5, cosec A = ______.
💡 Hint:
5/3.
Explanation:
Turn it over.
3
sin 90° = ______.
💡 Hint:
1.
Explanation:
The last cell of the table.
4
tan 30° = ______.
💡 Hint:
1/√3.
Explanation:
sin/cos.
5
cos²A = 1 − ______.
💡 Hint:
sin²A.
Explanation:
The first identity.
6
sec²A − tan²A = ______.
💡 Hint:
1.
Explanation:
The identity.
7
cos 0° = ______.
💡 Hint:
1.
Explanation:
The leftmost cos.
8
sin 45° = ______.
💡 Hint:
1/√2.
Explanation:
cos 45° is the same.
Match
0/21
Match the ratio with its meaning.
Column B: A. Hypotenuse/opposite · B. Opposite/hypotenuse · C. Adjacent/hypotenuse · D. Opposite/adjacent
1. sin A
2. cos A
3. tan A
4. cosec A
Explanation:
sin is opposite/hypotenuse, cos is adjacent/hypotenuse, tan is opposite/adjacent, and cosec is the reverse of sin.
2
Match the exercise with its job.
Column B: A. tan = 1 · B. Ratios from the sides · C. Values from 0° to 90° · D. sin²+cos²=1
1. 8.1
2. 8.2
3. 8.3
4. 45° in the table
Explanation:
8.1 is the ratios, 8.2 is the standard values, 8.3 is the identities. At 45°, tan = 1.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): If sin A = 3/5, then cos A = 4/5.
Reason (R): cos²A = 1 − sin²A and the root is positive for an acute angle.
Explanation:
Both are true and R is the reason.
2
Assertion (A): tan 45° = 1.
Reason (R): sin²A + cos²A = 1.
Explanation:
Both are true, but R does not explain this value. The value comes from the opposite and adjacent sides being equal.
3
Assertion (A): 9 sec²A − 9 tan²A = 9.
Reason (R): sec²A − tan²A = 0.
Explanation:
A is true. R is false, the difference is 1.
4
Assertion (A): sin θ = 4/3 is possible for some angle.
Reason (R): The opposite side cannot be longer than the hypotenuse.
Explanation:
A is false. R is true.
5
Assertion (A): sin 60° cos 30° + sin 30° cos 60° = 1.
Reason (R): Putting √3/2 and 1/2 leaves 3/4 + 1/4.
Explanation:
Both are true and R is the calculation.
Coefficient practice
0/4These chemistry lines are not a result of this maths chapter. They are only practice in filling coefficients. A blank means 1.
1
This is coefficient practice and not a result of this maths chapter. In the line where H2 + O2 makes water, Fill coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice and not a result of this maths chapter. In the line where N2 + H2 makes ammonia, Fill coefficients (blank = 1).
N2 + H2 → NH3
Explanation: N2 + 3H2 → 2NH3
3
This is coefficient practice and not a result of this maths chapter. In the line where C + O2 makes carbon dioxide, Fill coefficients (blank = 1).
C + O2 → CO2
Explanation: C + O2 → CO2
4
This is coefficient practice and not a result of this maths chapter. In the burning line of CH4 + O2, Fill coefficients (blank = 1).
CH4 + O2 → CO2 + H2O
Explanation: CH4 + 2O2 → CO2 + 2H2O
Classify
0/21
Place each statement as possible, impossible, or not defined.
sin θ = 4/3
sec A = 13/5
tan 90°
sin 30° = 1/2
Explanation:
4/3 is impossible. 13/5 is possible. tan 90° does not exist. sin 30° is 1/2 in the table.
2
Place each identity in its form.
Divide by the square of the hypotenuse
Divide by the square of the adjacent side
Divide by the square of the opposite side
9 sec²A − 9 tan²A
Explanation:
The hypotenuse gives the first identity, the adjacent side the sec form, the opposite side the cosec form. Nine times the difference is 9.
Very short answer
0/51
Write the definitions of sin A, cos A and tan A.
Model answer (board format)
sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
Check your answer:
2
Write sin 30°, sin 45° and sin 60°.
Model answer (board format)
1/2, 1/√2 and √3/2.
Check your answer:
3
Write the three identities.
Model answer (board format)
sin²A + cos²A = 1, 1 + tan²A = sec²A, 1 + cot²A = cosec²A.
Check your answer:
4
Write cosec A in terms of sin A.
Model answer (board format)
cosec A = 1/sin A.
Check your answer:
5
Why is sin θ = 4/3 impossible?
Model answer (board format)
The opposite side cannot be longer than the hypotenuse, so sin is never greater than 1.
Check your answer:
Short answer
0/41
The right angle is at B, AB = 5 cm, BC = 12 cm. Find the hypotenuse and sin A.
Model answer (board format)
Hypotenuse √(25+144) = √169 = 13 cm. sin A = opposite/hypotenuse = 12/13.
Check your answer:
2
Evaluate cos 30° sin 30° + sin 60° cos 60°.
Model answer (board format)
(√3/2)(1/2) + (√3/2)(1/2) = √3/4 + √3/4 = √3/2.
Check your answer:
3
If cot A = 3/4, find sin A.
Model answer (board format)
Adjacent 3, opposite 4, hypotenuse 5. sin A = 4/5.
Check your answer:
4
Write sec²A − tan²A and 1 + cot²A from the identities.
Model answer (board format)
sec²A − tan²A = 1. 1 + cot²A = cosec²A.
Check your answer:
Long answer
0/31
If tan A = 5/12 and A is acute, find sin A, cos A and sec A. For a check, show sin²+cos².
Model answer (board format)
Adjacent 12, opposite 5, hypotenuse √(144+25) = 13. sin A = 5/13, cos A = 12/13, sec A = 13/12. (25+144)/169 = 1.
Check your answer:
2
Prove that (sec A + tan A)(1 − sin A) = cos A, where the terms exist.
Model answer (board format)
sec A + tan A = (1 + sin A)/cos A. Multiply by (1 − sin A): (1 − sin²A)/cos A = cos²A / cos A = cos A.
Check your answer:
3
In a right triangle one acute angle has adjacent side 9 cm and hypotenuse 15 cm. Find sin, cos and tan, and name the Pythagorean triple.
Model answer (board format)
Opposite √(225−81) = √144 = 12 cm. sin = 12/15 = 4/5, cos = 9/15 = 3/5, tan = 12/9 = 4/3. The sides are 9, 12, 15, which is three times 3-4-5.
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BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination.
1
In the spirit of exercise 8.1, this is impossible —
💡 Hint:
sin is at most 1.
Explanation:
sin θ = 4/3 is impossible.
2
cos 0° + sin 90° equals —
💡 Hint:
1 + 1.
Explanation:
2.
3
tan 60° = ______.
💡 Hint:
√3.
Explanation:
sin/cos = (√3/2)/(1/2).
4
If cot A = 5/12, find sec A.
Model answer (board format)
Adjacent 5, opposite 12, hypotenuse 13. sec A = hypotenuse/adjacent = 13/5.
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5
Derive sin²A + cos²A = 1 from a right triangle in two lines. Then if cos A = 7/25, find sin A.
Model answer (board format)
opposite² + adjacent² = hypotenuse². Divide by hypotenuse²: sin²A + cos²A = 1. sin²A = 1 − 49/625 = 576/625. sin A = 24/25.
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6
Assertion (A): From 0° to 90°, sin increases.
Reason (R): In the same interval cos also increases.
Explanation:
A is true, from 0 to 1. R is false, cos falls from 1 to 0.
CBSE-style questions
0/6These are competency-based practice questions. They are not a copy of any year’s paper.
1
A student writes sin 60° + sin 30° as 1. The right statement is —
💡 Hint:
Keep the values apart.
Explanation:
√3/2 + 1/2. The product form sin 60 cos 30 + sin 30 cos 60 is the one that gives 1.
2
For an angle with adjacent side 9 cm and hypotenuse 15 cm, sin is —
💡 Hint:
Opposite 12.
Explanation:
sin = 12/15 = 4/5.
3
Assertion (A): 9 sec²A − 9 tan²A = 9.
Reason (R): sec²A − tan²A = 1.
Explanation:
Both are true and R is the basis of the factor 9.
4
Assertion (A): cos A is the short name of cosecant.
Reason (R): cosec A = 1/sin A and sec A = 1/cos A.
Explanation:
A is false. R is true.
5
A ramp makes 30° with the ground and is 4 m long. Find the height. Which ratio did you use?
Model answer (board format)
The height is opposite and the ramp is the hypotenuse. sin 30° = h/4. h = 4 × 1/2 = 2 m.
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6
If tan A = 8/15, write sin A and cos A as fractions.
Model answer (board format)
Opposite 8, adjacent 15, hypotenuse 17. sin A = 8/17, cos A = 15/17.
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CBSE-style · competency-based
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🧠 What you learned + equation sheet
What you learned
- sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
- cosec, sec and cot are the reciprocals of those.
- sin and cos are never greater than 1. sin θ = 4/3 is impossible.
- The sin row is 0, 1/2, 1/√2, √3/2, 1. The cos row is the reverse.
- Do not split sin(A + B) into sin A + sin B.
- sin²A + cos²A = 1, and sec²A − tan²A = 1.
| What | Keep this |
|---|---|
| sin A | opposite / hypotenuse |
| cos A | adjacent / hypotenuse |
| tan A | opposite / adjacent |
| sin 30° | 1/2 |
| tan 45° | 1 |
| Identity | sin²A + cos²A = 1 |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.