Question

question 7 (5 points)
listen
if ( sec \theta=-7.3 ), find ( sin left(\theta-\frac{pi}{2}
ight) ).
0.14
7.3
-0.14
-7.3
question 8 (5 points)
listen
solve ( 4+2 sin x=14-8 sin x ) for ( 0^{circ} leq x leq 180^{circ} ).
( 0^{circ} )
( 60^{circ} )
( 90^{circ} )
( 45^{circ} )

Explanation:

Question 7

Step1: Use trigonometric identity

We know that \(\sin(A - B)=\sin A\cos B-\cos A\sin B\). For \(\sin(\theta-\frac{\pi}{2})\), \(A = \theta\), \(B=\frac{\pi}{2}\). So \(\sin(\theta-\frac{\pi}{2})=\sin\theta\cos\frac{\pi}{2}-\cos\theta\sin\frac{\pi}{2}\). Since \(\cos\frac{\pi}{2} = 0\) and \(\sin\frac{\pi}{2}=1\), then \(\sin(\theta - \frac{\pi}{2})=-\cos\theta\).

Step2: Find \(\cos\theta\) from \(\sec\theta\)

Given \(\sec\theta=- 7.3\), and \(\sec\theta=\frac{1}{\cos\theta}\), so \(\cos\theta=\frac{1}{\sec\theta}=-\frac{1}{7.3}\approx - 0.14\).

Step3: Calculate \(\sin(\theta-\frac{\pi}{2})\)

Since \(\sin(\theta-\frac{\pi}{2})=-\cos\theta\), substituting \(\cos\theta\approx - 0.14\), we get \(\sin(\theta-\frac{\pi}{2})\approx0.14\)

Step1: Rearrange the equation

Starting with \(4 + 2\sin x=14-8\sin x\). Add \(8\sin x\) to both sides: \(4+2\sin x + 8\sin x=14-8\sin x+8\sin x\), which gives \(4 + 10\sin x=14\). Then subtract \(4\) from both sides: \(10\sin x=14 - 4=10\).

Step2: Solve for \(\sin x\)

Divide both sides of \(10\sin x = 10\) by \(10\), we get \(\sin x = 1\)

Step3: Find \(x\) in the given range

We know that \(\sin x = 1\) when \(x = 90^{\circ}\) for \(0^{\circ}\leq x\leq180^{\circ}\)

Answer:

\(0.14\)

Question 8