Introduction to Trigonometry — Class 10 MCQs with Answers

Introduction to Trigonometry — important questions & MCQs with answers (Class 10 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   In a right triangle, sin θ is defined as:

   AOpposite / HypotenuseBAdjacent / HypotenuseCOpposite / AdjacentDHypotenuse / Opposite
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   Answer: Opposite / Hypotenuse

   Explanation: By definition, sinθ = opposite side ÷ hypotenuse in a right triangle. It is easy to mix up with cosθ = adjacent ÷ hypotenuse — SOH-CAH-TOA keeps the pairing straight.

2. Q2 of 6Easy

   The value of sin 30° is:

   A√3/2B1/2C1D1/√2
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   Answer: 1/2

   Explanation: sin30° is a standard-angle value = 1/2, read off the 0°-30°-45°-60°-90° table. It is easy to confuse with cos30° = √3/2, since sin grows from 0 to 1 as the angle increases.

3. Q3 of 6Medium

   sin 30° + cos 60° equals:

   A√3B1/2C1D0
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   Answer: 1

   Explanation: sin30°=1/2, and by the co-function identity cos(90°−θ)=sinθ, cos60°=sin30°=1/2, so the sum is 1/2+1/2=1. Don't confuse cos60° with cos30°=√3/2.

4. Q4 of 6Medium

   Assertion (A): For the same acute angle A, the value of sin A does not change when the sides of the right triangle are made longer. Reason (R): tan A = sin A ÷ cos A.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: Both are true. A holds because the triangles with the same angle A are similar, so their sides are in the same ratio; R is a different true fact and does not explain A.

5. Q5 of 6Hard

   Value of: sin²5° + sin²10° + sin²15° + ... + sin²85° + sin²90°.

   A20/2B19/2C9D91/2
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   Answer: 19/2

   Explanation: The terms run 5°, 10°, …, 90° (18 terms). Since sin²θ + sin²(90° − θ) = 1, the eight pairs (5°,85°) … (40°,50°) contribute 8. Adding sin²45° = ½ and sin²90° = 1 gives 8 + ½ + 1 = 19/2.

6. Q6 of 6Hard

   Assertion (A): In triangle ABC, right-angled at B, if tan A = 1, then 2 sin A cos A = 1. Reason (R): If tan A = 1, then sin A = 1/2.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is true but R is false

   Explanation: A is true: tan A = 1 means AB = BC = k, so AC = √2 k and sin A = cos A = 1/√2, giving 2 sin A cos A = 1. R is false, because sin A = 1/√2, not 1/2.

Introduction to Trigonometry notes: key concepts (Class 10 Mathematics)

Trigonometric ratios attach numbers to the angles of a right triangle, so an unknown side or angle can be computed rather than measured. This chapter defines the six ratios, evaluates them at the standard angles, and establishes the fundamental identities.

Trigonometric ratios

The six ratios sin, cos, tan, cosec, sec and cot, each defined as a ratio of two sides of a right triangle relative to a chosen acute angle.

Opposite, adjacent and hypotenuse

Named relative to the angle in question: the hypotenuse is fixed, but which side is opposite and which adjacent changes with the angle chosen.

sin, cos and tan

sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent — the three from which all others follow.

Reciprocal ratios

cosec, sec and cot are the reciprocals of sin, cos and tan respectively, in that exact pairing.

tan in terms of sin and cos

tan θ = sin θ / cos θ, and cot θ = cos θ / sin θ — the link that converts between forms during simplification.

Standard angles

The exact values of the ratios at 0°, 30°, 45°, 60° and 90°, which must be known by heart for the board exam.

Range of the ratios

For an acute angle, sin and cos always lie between 0 and 1, so any answer outside that range signals an error.

Fundamental identity

sin²θ + cos²θ = 1, the Pythagoras theorem rewritten in terms of ratios, and the basis of nearly every simplification.

Secant and tangent identity

1 + tan²θ = sec²θ, obtained by dividing the fundamental identity through by cos²θ.

Cosecant and cotangent identity

1 + cot²θ = cosec²θ, obtained by dividing the fundamental identity through by sin²θ.

Complementary angles

sin(90° − θ) = cos θ and tan(90° − θ) = cot θ — each ratio turns into its co-ratio.

Proving identities

Work on one side alone, converting everything to sin and cos, until it matches the other side; never move terms across the equality.