Question

Please help me on my precalculus test.

Answer 1

14. Recall the double angle identity for sine:

sin(2x) = 2 sin(x) cos(x)

If x is in quadrant 1, then both sin(x) and cos(x) are positive, so

sin²(x) + cos²(x) = 1   →   cos(x) = + √(1 - sin²(x))

which gives cos(x) = 4/5. Then

sin(2x) = 2 • (3/5) • (4/5) = 24/25

15. Recall the double angle identity for cosine:

cos(2x) = cos²(x) - sin²(x)

which we can rewrite in terms of sin(x) only as

cos(2x) = 1 - 2 sin²(x)

Then cos(2x) = 1 - 2 • (3/5)² = 7/25

16. Recall the definition of tangent:

tan(x) = sin(x) / cos(x)

Then tan(2x) = sin(2x) / cos(2x) = (24/25) / (7/25) = 24/7