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Trigonometric Ratios Quiz - CBSE Class 10 Mathematics

Free practice quiz: Trigonometric Ratios (CBSE Class 10 Mathematics). 16 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 10 Mathematics practice quiz has 16 multiple-choice questions on Trigonometric Ratios. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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About this topic

In a right-angled triangle, the ratios of sides depend only on the acute angle, not on the triangle's size. There are six ratios: sin, cos, tan and their reciprocals cosec, sec, cot. Standard values at 0°, 30°, 45°, 60°, 90° and the identities link them.

For acute angle θ: Opposite = side facing θ; Adjacent = side touching θ (not hypotenuse).
Hypotenuse is always the longest side, opposite the right angle.
sin θ and cos θ are never more than 1; sec θ and cosec θ are never less than 1.
Ratios stay same for similar triangles, so they depend only on the angle.
Complementary angles: sin (90° − θ) = cos θ and tan (90° − θ) = cot θ.
Given one ratio, draw a right triangle, find the third side by Pythagoras, then write all ratios.
tan θ and sec θ are undefined at 90°; cot θ and cosec θ are undefined at 0°.

Learn the key ideas Beta

The six trigonometric ratios

In a right triangle, each ratio compares two sides, named from the chosen acute angle A.
Opposite faces angle A, adjacent touches angle A, and the hypotenuse faces the right angle.
sec, cosec and cot are reciprocals of cos, sin and tan respectively.

Sine and cosine

\[\sin A = \frac{\text{opposite}}{\text{hypotenuse}},\quad \cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\]

Tangent

\[\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin A}{\cos A}\]

Reciprocal ratios

\[\sec A = \frac{1}{\cos A},\quad \csc A = \frac{1}{\sin A},\quad \cot A = \frac{1}{\tan A}\]

Common mistake: Writing sec A as 1/sin A. Remember sec pairs with cos, and cosec pairs with sin.

Finding standard angle values

Do not just memorise. Build special triangles and read the ratios from their sides.
Halve an equilateral triangle of side 2a, or draw an isosceles right triangle with legs a.
If A and B add up to a right angle, then sin A equals cos B.

Isosceles right triangle with legs a

\[\text{hypotenuse} = a\sqrt{2}\]

Equilateral triangle with side 2a

\[\text{altitude} = a\sqrt{3},\ \text{half-base} = a\]

Complementary angles

\[A + B = 90^\circ \Rightarrow \sin A = \cos B,\ \tan A = \cot B\]

Common mistake: Mixing up opposite and adjacent sides in the half-equilateral triangle. Always relabel the sides from the angle you are using.

Pythagorean identities

These come from Pythagoras' theorem and hold for every angle A.
If one ratio is known, use them to find the others without drawing the triangle.

Basic identity

\[\sin^2 A + \cos^2 A = 1\]

Rearranged forms

\[\cos^2 A = 1 - \sin^2 A,\quad \sin^2 A = 1 - \cos^2 A\]

With tan and sec

\[1 + \tan^2 A = \sec^2 A\]

With cot and cosec

\[1 + \cot^2 A = \csc^2 A\]

Common mistake: Writing cos A = 1 − sin A. You must square sin A, subtract it from 1, then take the square root.

Angles beyond acute and radians

This goes beyond the NCERT Class 10 syllabus, but some quiz questions need it.
The sign of each ratio depends on the quadrant of the angle.
Radians are another unit for measuring angles.

Degrees to radians

\[\pi \text{ rad} = 180^\circ\]

Supplementary angle

\[\sin(180^\circ - A) = \sin A,\quad \cos(180^\circ - A) = -\cos A\]

Positive ratios by quadrant (ASTC)

\[\text{I: all},\ \text{II: } \sin,\ \text{III: } \tan,\ \text{IV: } \cos\]

Common mistake: Taking the positive square root every time. Check the quadrant first, then choose the correct sign.

Exam tips

Draw a quick right triangle and label the opposite, adjacent and hypotenuse before choosing any ratio.
After using an identity, check that sin and cos lie between −1 and 1.
Look for Pythagorean triples like 5, 12, 13 to find the missing side quickly.

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

What is the value of sin(30°)?
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Answer: B. 0.5
What is the value of sin(60°)?
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Answer: A. √3/2
What is the sin of 90°?
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Answer: A. 1
What is the relationship between sin(θ) and cos(θ) for θ = 30°?
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Answer: A. sin(30°) = cos(60°)
If sin(θ) = 0.5, what is the value of θ in radians?
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Answer: A. π/6