MathIsimple

Trigonometric Identities

10 problems covering sum/difference formulas, double-angle formulas, and auxiliary angle techniques

1
Tangent Sum Formula Application

Problem:

Given that tan⁡α=2\tan\alpha = 2 and 2sin⁡α=cos⁡(α−β)sin⁡β2\sin\alpha = \cos(\alpha-\beta)\sin\beta, find tan⁡β\tan\beta.

2
Cosine Sum Formula

Problem:

Evaluate cos⁡147°cos⁡333°+cos⁡57°cos⁡63°\cos147°\cos333° + \cos57°\cos63°.

3
Roots and Tangent Formula

Problem:

Given that tan⁡α\tan\alpha and tan⁡β\tan\beta are roots of x2−7x+13=0x^2 - 7x + 13 = 0, find tan⁡(α+β)\tan(\alpha+\beta).

4
Sine Sum Formula Application

Problem:

Given sin⁡(2α−β)=513\sin(2\alpha-\beta) = \frac{5}{13} and cos⁡(α−β)=13sin⁡α\cos(\alpha-\beta) = \frac{1}{3}\sin\alpha, find sin⁡α\sin\alpha.

5
Sum of Squares Method

Problem:

Given cos⁡α+cos⁡β=1010\cos\alpha + \cos\beta = \frac{\sqrt{10}}{10} and sin⁡α+sin⁡β=31010\sin\alpha + \sin\beta = \frac{3\sqrt{10}}{10}, find cos⁡(α−β)\cos(\alpha-\beta).

6
Double-Angle Formula

Problem:

Given sin⁡α−cos⁡α=13\sin\alpha - \cos\alpha = \frac{1}{3} and α∈(0,π)\alpha \in (0,\pi), find cos⁡2α\cos 2\alpha.

7
Reduction and Double-Angle

Problem:

Given sin⁡(α−π6)=14\sin\left(\alpha - \frac{\pi}{6}\right) = \frac{1}{4}, find sin⁡(2α+5π6)\sin\left(2\alpha + \frac{5\pi}{6}\right).

8
Angle Evaluation

Problem:

Evaluate 1+3tan⁡10°1+cos⁡20°\frac{1 + \sqrt{3}\tan 10°}{1 + \cos 20°}.

9
Auxiliary Angle Method

Problem:

Given 2sin⁡α+1=23cos⁡α2\sin\alpha + 1 = 2\sqrt{3}\cos\alpha, find sin⁡(2α+π6)\sin\left(2\alpha + \frac{\pi}{6}\right).

10
Value Finding Comprehensive

Problem:

Given sin⁡α+sin⁡(α+π3)=33\sin\alpha + \sin\left(\alpha + \frac{\pi}{3}\right) = \frac{\sqrt{3}}{3}, find cos⁡(2α+π3)\cos\left(2\alpha + \frac{\pi}{3}\right).