91Ó°ÊÓ

One way to solve an equation with a graphing calculator is to rewrite the equation with 0 on the right-hand side, then graph the function that is on the left-hand side. The x-coordinate of each \(x\) -intercept of the graph is a solution to the original equation. For each equation, find all real solutions (to the nearest tenth) in the interval \([0,2 \pi).\) $$x^{2}=\sin x$$

$$\text { Use identities to simplify } \frac{1}{\cos ^{2} x}-\tan ^{2} x$$

Use identities to simplify the expression \(\frac{\csc x}{\sec x}\)

Motion of a Spring A block is attached to a spring and set in motion on a frictionless plane. Its location on the surface at any time \(t\) in seconds is given in meters by \(x=\sqrt{3} \sin 2 t+\cos 2 t .\) For what values of \(t\) is the block at its resting position \(x=0 ?\)

Suppose that \(\sin \alpha=1 / 4\) and \(\alpha\) is in quadrant II. Use identities to find the exact values of the other five trigonometric functions.

Solve each problem. Motion of a Spring A block is set in motion hanging from a spring and oscillates about its resting position \(x=0\) according to the function \(x=-0.3 \sin 3 t+0.5 \cos 3 t\). For what values of \(t\) is the block at its resting position \(x=0 ?\)

Find the point that lies midway between \((\pi / 3,1)\) and \((\pi / 2,1)\)

Solve each problem. Wave Action The vertical position of a floating ball in an experimental wave tank is given by the equation \(x=2 \sin (\pi t / 3)\) where \(x\) is the number of feet above sea level and \(t\) is the time in seconds. For what values of \(t\) is the ball \(\sqrt{3} \mathrm{ft}\) above sea level?

Determine the amplitude, period, and phase shift for the function \(y=-4 \sin (2 \pi x / 3-\pi / 3)\)

Trigonometric Identities List as many trigonometric identities as you can and explain why each one is an identity.