Trigonometry - Sine and Cosine Functions help

If sin d° = 0.25 and sin k° = 0.75, which of the following is the value of cos d° + cos k°?

a. 1.68

b. 1.59

c. 1.61

d. 1.63


If sin m° = 0.5, which of the following is the value of cos m°?

a. 0.85

b. 0.87

c. 0.78

d. 0.67

cos(x)^2 + sin(x)^2 = 1

cos(x)^2 = 1 - sin(x)^2
cos(x) = ±√(1 - sin(x)^2)
sin(d) = .25 => cos(d) = ±√(1 - .0625) = ±√.9375
sin(k) = .75 => cos(k) = ±√(1 - .5625) = ±√.4375
So cos(d) + cos(k) ≈ 1.63, if d and k are both in the first quadrant.

sin(m) = .5 => cos(m) = ±√(1 - .25) = ±√.75
Assuming m is in the first quadrant, cos(m) ≈ 0.87.

P = perp = opp.side, B = base = adj.side and H = hypot.
sin d = P / H = 0.25 = 1/4 = m / 4m where m is a constant
=> P = m and H = 4m
=> B = √(H² - P²) = √15m
=> cos d = B / H = √15m / 4m = √15 / 4
Similarly,
sin k = P / H = 0.75 = 3/4 = 3m / 4m
cos k = B / H = √(H² - P²) / H = √7m / 4m = √7 / 4
cos d + cos k = (√15 / 4) + (√7 / 4)
= (√15 + √7) / 4 = (3.87 + 2.64) / 4 = 6.51 / 4 = 1.63 ........ (d)
Try the 2nd question in the same manner yourself.