If sin 23=p,write the following in terms of p cos 113,cos 23,tan 23

recall that cos(90+x) = -sin(x)

sin^2 + cos^2 = 1
tan = sin/cos

-sin 23

To express cos 113, cos 23, and tan 23 in terms of p, we need to use the trigonometric identities and make use of the given relationship sin 23 = p.

1. cos 23:
We can use the Pythagorean identity to find cos 23.
sin^2(23) + cos^2(23) = 1

Since sin 23 = p, we can substitute:
p^2 + cos^2(23) = 1

Solving for cos 23:
cos^2(23) = 1 - p^2
cos 23 = ±√(1 - p^2)

2. cos 113:
We can use the identity cos(180 - θ) = -cos(θ) to find cos 113.
cos 113 = -cos(180 - 113)
cos 113 = -cos 67

Since cos 67 = sin(90 - 67) = sin 23 = p (given), we can substitute:
cos 113 = -p

3. tan 23:
We can use the identity tan θ = sin θ / cos θ to find tan 23.
tan 23 = sin 23 / cos 23

Since sin 23 = p and cos 23 = ±√(1 - p^2) (as found earlier), we can substitute:
tan 23 = p / ±√(1 - p^2)

Therefore, in terms of p, the expressions are:
cos 113 = -p
cos 23 = ±√(1 - p^2)
tan 23 = p / ±√(1 - p^2)