Exam 6: Trigonometric Functions: Right Triangle Approach

Find the angle $\theta$ in the triangle.

Two cities in the western hemisphere lie approximately on the same meridian. Guatemala city has latitude of $15.5 ^ { \circ }$ N and Baton Rouge, Louisiana $30.5 ^ { \circ }$ N. Find the distance between these two cities. (The radius of the earth is $3960$ miles.)

Find the length of an arc of a circle of radius $8$ m if the arc subtends a central angle of $5 \pi / 24$ .

Find the area of the shaded region.

The time in seconds that it takes for a sled to slide down a hillside inclined at an angle $\theta$ is $t = \sqrt { \frac { d } { 16 \sin \theta } }$ where d is the length of the slope in feet. Find the time it takes to slide down a 4000-ft slope inclined at $30 ^ { \circ }$

A 76-ft tree casts a shadow that is 100 ft long. What is the angle of elevation of the sun?

A surveyor stands at a point $1.5$ miles away from the base of a radio tower. He measures the angle of elevation to the top of the tower to be $7.5 ^ { \circ }$ . Find the height of the tower to the nearest foot.

A $200$ ft guy wire is attached to the top of a tower. If the wire makes a $55 ^ { \circ }$ angle with the ground, how tall is the tower?

Find the area of a triangle with sides of length 12 and 22 and included angle $12 ^ { \circ }$ .

A helicopter is dropping water on a forest fire from a height of $300$ ft. If one side of the fire makes an angle of depression of $40 ^ { \circ }$ , and the other side makes an angle of depression of $85 ^ { \circ }$ on the other side of the helicopter, how wide is the fire.

Use the Law of Cosines to find $x$ .

Find the exact value of each expression, if it is defined. (a) $\cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right)$ (b) $\sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right)$ (c) $\tan ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 3 } \right)$

Find the area of a sector with central angle $3 \pi / 2$ in a circle with radius $5$ m.

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $a = 101 , b = 81 , \quad \angle A = 135 ^ { \circ }$

Find the exact value of $\tan ( - 8 \pi / 3 )$ .

Solve the triangle. $\angle A = 23 ^ { \circ }$ , $b = 78$ , $c = 88$

Find the exact value of the expression. $\sec \left( \sin ^ { - 1 } \frac { 5 } { 13 } \right)$

Find the reference angle for $\theta = \frac { 7 \pi } { 6 }$ .

A $50$ ft tree casts a shadow that is $100$ ft long. What is the angle of elevation of the sun?

Find the exact value of $\cos ( 6 \pi / 4 )$ .