Question

find an angle \\( \theta \\) with \\( 0^{\circ}<\theta<360^{\circ} \\) that has the same:
sine as \\( 10^{\circ}: \theta= \\) degrees
cosine as \\( 10^{\circ}: \theta= \\) degrees
question help: video

Explanation:

Step1: Find the angle with the same sine as \(10^{\circ}\)

Recall the identity \(\sin\theta=\sin(180^{\circ}-\theta)\).
For \(\theta = 10^{\circ}\), the angle with the same sine in the range \(0^{\circ}<\theta<360^{\circ}\) is \(180^{\circ}- 10^{\circ}=170^{\circ}\) since \(\sin10^{\circ}=\sin170^{\circ}\) (sine is positive in the first and second quadrants).

Step2: Find the angle with the same cosine as \(10^{\circ}\)

Recall the identity \(\cos\theta=\cos(360^{\circ}-\theta)\).
For \(\theta = 10^{\circ}\), the angle with the same cosine in the range \(0^{\circ}<\theta<360^{\circ}\) is \(360^{\circ}-10^{\circ} = 350^{\circ}\) since \(\cos10^{\circ}=\cos350^{\circ}\) (cosine is positive in the first and fourth quadrants).

Answer:

Sine as \(10^{\circ}\): \(\theta = 170\) degrees
Cosine as \(10^{\circ}\): \(\theta=350\) degrees