Question

1. if $\sin\theta =.6000$, find the value of $\sin(180^{\circ}+\theta)$. 21. if $\cos25^{\circ}=.9063$, find $\cos155^{\circ}$.

Explanation:

Step1: Use trigonometric identity

For \( \sin(180^{\circ}+\theta)\), use the identity \( \sin(A + B)=\sin A\cos B+\cos A\sin B\). Here \(A = 180^{\circ}\), \(B=\theta\). So \( \sin(180^{\circ}+\theta)=\sin180^{\circ}\cos\theta+\cos180^{\circ}\sin\theta\). Since \( \sin180^{\circ}=0\) and \( \cos180^{\circ}=- 1\), then \( \sin(180^{\circ}+\theta)=-\sin\theta\).
Given \( \sin\theta = 0.6000\), so \( \sin(180^{\circ}+\theta)=-0.6000\).

Step2: For \( \cos155^{\circ}\)

Use the identity \( \cos(A + B)=\cos A\cos B-\sin A\sin B\). We can write \(155^{\circ}=180^{\circ}-25^{\circ}\). So \( \cos155^{\circ}=\cos(180^{\circ}-25^{\circ})\). Using the identity \( \cos(A - B)=\cos A\cos B+\sin A\sin B\) with \(A = 180^{\circ}\), \(B = 25^{\circ}\), we get \( \cos(180^{\circ}-25^{\circ})=\cos180^{\circ}\cos25^{\circ}+\sin180^{\circ}\sin25^{\circ}\). Since \( \sin180^{\circ}=0\) and \( \cos180^{\circ}=-1\), then \( \cos155^{\circ}=-\cos25^{\circ}\). Given \( \cos25^{\circ}=0.9063\), so \( \cos155^{\circ}=- 0.9063\).

Answer:

For \( \sin(180^{\circ}+\theta)\), the value is \(-0.6000\). For \( \cos155^{\circ}\), the value is \(-0.9063\).