IMO Practice Test: Trigonometry | Class 10 Mathematics

Question 1 of 6 medium

If \(\sin \theta\) + \(\sin^{2}\theta\) = 1, find the value of the expression: \(\cos^{2}\theta\) + \(\cos^{4}\theta\).

0

1

2

1 / 2

Explanation: \(\sin \theta\) = 1 - \(\sin^{2}\theta\) = \(\cos^{2}\theta\). Squaring gives \(\sin^{2}\theta\) = \(\cos^{4}\theta\). Thus cos^2 + cos^4 = cos^2 + sin^2 = 1.

Question 2 of 6 medium

Find the numerical value of the following sequence: log(tan 1°) + log(tan 2°) + ... + log(tan 89°).

1

0

-1

Undefined

Explanation: log sum = log(tan 1 * ... * tan 89). Complementary terms pair to 1 (tan 1 * tan 89 = 1), tan 45 = 1. log(1) = 0.

Question 3 of 6 medium

Eliminate theta from the given equations: x = a * \(\sec \theta\) and y = b * \(\tan \theta\). Which relation is correct?

x^2/a^2 + y^2/b^2 = 1

x^2/a^2 - y^2/b^2 = 1

x^2 + y^2 = a^2 + b^2

y^2/b^2 - x^2/a^2 = 1

Explanation: \(\sec \theta\) = x/a, \(\tan \theta\) = y/b. Apply identity sec^2 - tan^2 = 1 to get x^2/a^2 - y^2/b^2 = 1.

Question 4 of 6 medium

If \(\sec \theta\) + \(\tan \theta\) = p, find the value of \(\cosec \theta\) expressed in terms of p.

(p^2 + 1)/(p^2 - 1)

(p^2 - 1)/(p^2 + 1)

2p/(p^2 - 1)

(p^2 + 1)/2p

Explanation: sec - tan = 1/p. Adding gives 2sec = p + 1/p. Subtracting gives 2tan = p - 1/p. cosec = sec/tan = (p^2+1)/(p^2-1).

Question 5 of 6 medium

Find the value of the compound numeric expression: 2 * (\(\sin^{6}\theta\) + \(\cos^{6}\theta\)) - 3 * (\(\sin^{4}\theta\) + \(\cos^{4}\theta\)) + 1.

1

0

-1

2

Explanation: Expanding using identities simplifies this expression down to a constant 0 for any angle theta.

Question 6 of 6 medium

If A, B, and C are the interior angles of a triangle ABC, which of the following relations is mathematically correct?

sin((B+C)/2) = cos(A/2)

tan((B+C)/2) = tan(A/2)

cos((B+C)/2) = sin(A)

sin(B+C) = cos(A)

Explanation: B+C = 180 - A. Dividing by 2 gives (B+C)/2 = 90 - A/2. sin(90 - A/2) = cos(A/2).

IMO Practice Test — Trigonometry