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Exam 7: Trigonometry and Periodic Functions

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Question 1

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If the $y$ -value for the point on the unit circle with angle $\alpha^{\circ}$ is 0.39 , find $\cos \alpha$ and $\sin \alpha$ . Round your answers to three decimal places, if necessary.

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Question 2

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Consider the following figure. If $A=90^{\circ}, \sin C=0.49$ , and $c=7$ , find $a$ and $b$ . Round your answers to 2 decimal places, if necessary.

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Question 3

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A wheel has radius 6 inches and spins at a rate of 766 degrees per second. A red dot is painted on the outermost part of the wheel. How far (inches) has the dot traveled in 7 seconds? Round your answer to 3 decimal places.

In the following figure, the coordinates of $Q$ are $(0.28,-0.96)$ . The coordinates of $\mathrm{P}$ are ( --------------,-----------). Round to 2 decimal places.

What is the horizontal shift of $y=7 \cos (2 t+6)-6$ ? Round to 2 decimal places.

Two ships start at the same port. Ship A travels due east at $10 \mathrm{mph}$ . Ship B travels due north. After 3 hours, the ships are 56.60 miles apart. How fast was ship B traveling? Round your answer to the nearest whole number.

Solve for $\theta$ , an angle in a right triangle, if $5 \cos (2 \theta)+7=\cos (2 \theta)+8$ . Give the answer correct to 3 decimal places.

Find the following exactly using the figure given if $a=3$ and $c=7$ . Express your answers as unsimplified radicals when appropriate. A) $\sin A^{\circ}$ B) $\cos A^{\circ}$ C) $\tan A^{\circ}$ D) $\sin B^{\circ}$ E) $\cos B^{\circ}$ F) $\tan B^{\circ}$

Does the following function appear to be periodic with period $\leq 4$ ?

The midline of the periodic function $y=2 \cos (2 x)-4$ is $y=$ --------------

Point A lies on a circle of radius 2.5 at angle $251^{\circ}$ . Point $\mathrm{B}$ lies on a circle of radius 3 at angle $267^{\circ}$ . If both circles are centered at the origin, which point has the least $x$ -value?

Which of the following graphs have period 5 and amplitude 2 ?

City $A$ is a modest tourist town, which means that its population undergoes a seasonal variation. In January, it dips down to 4,000 people. By July, with the warm weather, its population climbs to around 5,000 people. This trend repeats every year. City $B$ , on the other hand, is a small town not far from City $A$ . Its population has been growing extremely rapidly ever since the arrival of several large industrial plants. There were only 3,500 people living in City $B$ on January 1, 2004, but its population has been growing by $10 \%$ every year thereafter. At how many different points in time will the population of City $A$ equal the population of City $B$ ?

What is the amplitude of the periodic function shown below?

Without a calculator, find the exact value of $\sec 240^{\circ}$ . If it is undefined, enter "undefined".

Estimate the period, midline, and amplitude of the periodic function with the following graph :

In the revolving door in the figure below, each panel is 1.1 meters long. How many meters is the arc length between $A$ and $D$ ? Round to 2 decimal places.

Find $\theta$ , an angle in a right triangle, if $\cos \theta=0.304$ . Give your answer to 3 decimal places.

An ant walks up a ramp at an incline of $11^{\circ}$ . If, after 2 minutes, his vertical distance from the ground is 13 inches, how fast was the ant travelling in inches per minute? Round your answer to 2 decimal places.

Using the identity $\sin t=\cos \left(t-\frac{\pi}{2}\right)$ find an angle $\theta$ such that $\sin \left(\frac{3 \pi}{4}\right)=\cos \theta$ .