Question

1. let $\cos(60)=\frac{1}{2}$. find the angle measure $(\theta)$ in degrees, for $\sin(\theta)=\frac{1}{2}$. 7. let $\sin(60)=\frac{\sqrt{3}}{2}$. find the angle measure $(\theta)$ in degrees, for $\cos(\theta)=\frac{\sqrt{3}}{2}$. 8. let $\cos(45)=\frac{\sqrt{2}}{2}$. find the angle measure $(\theta)$ in degrees, for $\sin(\theta)=\frac{\sqrt{2}}{2}$. 9. let $\cos(25)=0.99$. find the angle measure $(\theta)$ in degrees, for $\sin(\theta)=0.99$.

Explanation:

Step1: Recall trigonometric identities

We know that \(\sin^{2}\theta+\cos^{2}\theta = 1\). But for some standard angles, we can directly use the known values.

Step2: Solve for problem 6

Given \(\cos(60)=\frac{1}{2}\) and we need to find \(\theta\) for \(\sin\theta=\frac{1}{2}\).
We know that \(\sin(30)=\frac{1}{2}\).

Step3: Solve for problem 7

Given \(\sin(60)=\frac{\sqrt{3}}{2}\) and we need to find \(\theta\) for \(\cos\theta=\frac{\sqrt{3}}{2}\).
We know that \(\cos(30)=\frac{\sqrt{3}}{2}\).

Step4: Solve for problem 8

Given \(\cos(45)=\frac{\sqrt{2}}{2}\) and we need to find \(\theta\) for \(\sin\theta=\frac{\sqrt{2}}{2}\).
We know that \(\sin(45)=\frac{\sqrt{2}}{2}\).

Step5: Solve for problem 9

Given \(\cos(25) = 0.99\) and we need to find \(\theta\) for \(\sin\theta=0.99\).
Since \(\sin(90 - \alpha)=\cos\alpha\), here \(\alpha = 25\), so \(\sin(65)\approx0.99\) (using \(\sin(90 - 25)=\cos(25)\))

Answer:

1. \(30\) degrees
2. \(30\) degrees
3. \(45\) degrees
4. \(65\) degrees