Mathematics

Mathematics

Problem

Two angles α\alpha and β\beta satisfy sin⁡α=−513\sin\alpha=-\dfrac{5}{13} with α\alpha in the third quadrant and cos⁡β=45\cos\beta=\dfrac45 with β\beta in the first quadrant. a. (i) Find cos⁡α\cos\alpha and sin⁡β\sin\beta. b. (ii) For each angle, verify that sin⁡2+cos⁡2=1\sin^{2}+\cos^{2}=1 holds true with the values obtained.
Intuitive explanation

The key is to find the missing trig values for each angle using the Pythagorean identity, then apply an angle-sum or difference formula. Since you know one ratio for each angle and their quadrants, you can determine the signs of the other ratios—this unlocks everything you need to compute combined expressions like sin⁡(α+β)\sin(\alpha+\beta) or cos⁡(α−β)\cos(\alpha-\beta).

Intuitive explanation

The key is to find the missing trig values for each angle using the Pythagorean identity, then apply an angle-sum or difference formula. Since you know one ratio for each angle and their quadrants, you can determine the signs of the other ratios—this unlocks everything you need to compute combined expressions like sin⁡(α+β)\sin(\alpha+\beta) or cos⁡(α−β)\cos(\alpha-\beta).

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