Question

If cos 23°=p , express, without the use of a calculator, the following in terms of p:

sin 293°

Answer 1

To express sin 293° in terms of p, when cos 23°=p, we use the fact that sin 293° is equivalent to -sin 67° and sin(90° - θ) = cos(θ) to find that sin 293° = -p.

Step-by-step explanation:

If cos 23°=p, we want to express sin 293° in terms of p without the use of a calculator. To find this relationship, we should consider the properties of trigonometric functions in different quadrants and the co-function identities.

Firstly, 293° is in the fourth quadrant where sine is negative, and its co-function, cosine is positive. The reference angle for 293° is 360° - 293° = 67°. Since cosine is positive in the fourth quadrant and sin(90° - θ) = cos(θ), we have:

sin 293° = -sin(360° - 293°) = -sin 67°

But, 67° is just 90° - 23°, so:

sin 67° = cos 23° = p (because sin(90° - θ) = cos(θ))

Therefore, sin 293° = -p.