Question 1

An observer for a radar station is located at the origin of a coordinate system. For the point given, find the bearing of an
airplane located at that point. Express the bearing using both methods.
-(0, )

Accepted Answer

The point is directly on the y-axis above the origin, indicating a due north direction, which corresponds to a bearing of $0^\circ$ or $N 0^\circ W$ / $N 0^\circ E$.

Question 2

Find the sign of the following.
-$\csc ( - \theta )$, given that $\theta$ is in the interval $\left( 90 ^ { \circ } , 180 ^ { \circ } \right)$.

Accepted Answer

The cosecant function, $\csc(\theta)$, is the reciprocal of the sine function. For angles in the interval $90^\circ < \theta < 180^\circ$, sine is positive because these angles fall in the second quadrant where sine values are positive. However, since the angle is negative $(- \theta)$, it reflects the angle to the fourth quadrant where sine values are negative. Since $\csc(\theta) = 1/\sin(\theta)$, $\csc(-\theta)$ will also be negative due to the negative sine value in the fourth quadrant.

Question 3

Find the sign of the following.
-cos (ϴ + 180°), given that ϴ is in the interval (90°, 180°).

Accepted Answer

cos (ϴ + 180°) = -cos ϴ, and cos ϴ is negative in the interval (90°, 180°).

Question 4

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
-cos 47° cos 133° - sin 47° sin 133°

Accepted Answer

This expression is an application of the cosine addition formula, cos(A + B) = cosA cosB - sinA sinB. Here, A = 47° and B = 133°, so the expression simplifies to cos(47° + 133°) = cos(180°), which equals -1.

Question 5

Determine whether the statement is true or false.
-sin 390° = sin 120° cos 270° + cos 120° sin 270°

Accepted Answer

This statement is true because it is an application of the sine addition formula: sin(A + B) = sinA cosB + cosA sinB, where A = 120° and B = 270°, resulting in sin(120° + 270°) = sin 390°.

Question 6

Solve the right triangle.
-A = 19° 17´, c = 287 ft, C = 90° Round side lengths to two decimal places, if necessary.

Accepted Answer

Angle B is 70° 43' (since B = 90° - A), a = c * sin(A) = 287 * sin(19° 17') = 94.78 ft, and b = c * cos(A) = 287 * cos(19° 17') = 270.9 ft.

Question 7

Solve the right triangle.
-a = 3.6 m, B = 30.3°, C = 90° Round values to one decimal place.

Accepted Answer

The answer of Solve the right triangle.
-a = 3.6 m,...

Question 8

Solve the problem.
-Any offset between a stationary radar gun and a moving target creates a "cosine effect" that reduces the radar mileage reading by the cosine of the angle between the gun and the vehicle. That is, the
Radar speed reading is the product of the actual reading and the cosine of the angle. Find the radar
Reading to the nearest hundredth for the auto shown in the figure.

Accepted Answer

The answer of Solve the problem.
-Any offset between a stationary...

Question 9

Find the sign of the following.
-$\tan \frac { \theta } { 2 }$, given that $\theta$ is in the interval $\left( 90 ^ { \circ } , 180 ^ { \circ } \right)$.

Accepted Answer

In the interval $ (90^\circ, 180^\circ) $, $\theta$ is in the second quadrant where sine is positive and cosine is negative. Since $\tan \frac{\theta}{2} = \frac{\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}}$, and both sine and cosine of $\frac{\theta}{2}$ will be positive in this interval (as $\frac{\theta}{2}$ will be in the first quadrant), the tangent of $\frac{\theta}{2}$ will be positive.

Question 10

Solve the right triangle. If two sides are given, give angles in degrees and minutes.
-
$\mathrm { a } = 10.9 \mathrm {~cm} , \mathrm {~b} = 21.7 \mathrm {~cm}$
Round the missing side length to one decimal place.

Accepted Answer

The answer of Solve the right triangle. If two sides...

Question 11

Solve the problem.
-The index of refraction for air, Ia, is 1.0003. The index of refraction for water, Iw, is 1.3. If $\frac { \mathrm { I } _ { \mathrm { W } } } { \mathrm { I } _ { \mathrm { a } } } = \frac { \sin \mathrm { A } } { \sin \mathrm { W } }$, and $\mathrm { A } = 31.5 ^ { \circ }$, find $\mathrm { W }$ to the nearest tenth.

Accepted Answer

The answer of Solve the problem.
-The index of refraction for...

Question 12

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using
the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. Rationalize
the denominator if applicable.
-Find $\tan B$ when $b = 8$ and $c = 9$.

Accepted Answer

First, use the Pythagorean theorem $a^2 + b^2 = c^2$ to find $a$. Given $b = 8$ and $c = 9$, we have $a^2 + 8^2 = 9^2$, which simplifies to $a^2 + 64 = 81$. Solving for $a$ gives $a^2 = 17$, so $a = \sqrt{17}$. Then, $\tan B = \frac{opposite}{adjacent} = \frac{a}{b} = \frac{\sqrt{17}}{8}$. However, since $\tan B$ is actually $\frac{opposite}{adjacent}$ from the perspective of angle B, and the opposite side is $b = 8$ and the adjacent side is $a = \sqrt{17}$, the correct expression for $\tan B$ is $\frac{8}{\sqrt{17}}$, which, when rationalized, is $\frac{8\sqrt{17}}{17}$.

Question 13

Find a solution for the equation. Assume that all angles are acute angles.
-tan(3ϴ + 16°)= cot(ϴ + 4°)

Accepted Answer

The answer of Find a solution for the equation. Assume...

Question 14

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
-cos 39° 5' cos 50° 55' - sin 39° 5' sin 50° 55'

Accepted Answer

The answer of Use a calculator to find the function...

Question 15

Solve the right triangle. If two sides are given, give angles in degrees and minutes.
-

$\mathrm { A } = 41.3 ^ { \circ } , \mathrm { b } = 2.5 \mathrm {~m}$
Round side lengths to one decimal place.

Accepted Answer

The answer of Solve the right triangle. If two sides...

Question 16

Find a value of in [0°, 90°] that satisfies the statement. Leave answer in decimal degrees rounded to seven decimal
places, if necessary.
-csc ϴ = 1.7437345

Accepted Answer

The cosecant function is the reciprocal of the sine function. Therefore, csc ϴ = 1.7437345 implies sin ϴ = 1/1.7437345. Calculating this gives sin ϴ ≈ 0.5735764, which corresponds to an angle of approximately 34.9933809° in the range [0°, 90°].

Question 17

Determine whether the statement is true or false.
-cos 60° = cos 180° - cos 120°

Accepted Answer

cos 60° = 1/2, while cos 180° - cos 120° = -1 - (-1/2) = -1/2, so the two values are not equal.

Question 18

Solve the problem.
-Any offset between a stationary radar gun and a moving target creates a "cosine effect" that reduces the radar mileage reading by the cosine of the angle between the gun and the vehicle. That is, the
Radar speed reading is the product of the actual reading and the cosine of the angle. Find the radar
Reading to the nearest hundredth for the auto shown in the figure.

Accepted Answer

The answer of Solve the problem.
-Any offset between a stationary...

Question 19

Solve the problem.
-From a balloon 1037 feet high, the angle of depression to the ranger headquarters is 58°41'. How far is the headquarters from a point on the ground directly below the balloon (to the nearest foot)?

Accepted Answer

The answer of Solve the problem.
-From a balloon 1037 feet...

Question 20

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using
the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. Rationalize
the denominator if applicable.
-Find $\sec B$ when $a = 2$ and $b = 9$.

Accepted Answer

The answer of Suppose ABC is a right triangle with...

Quiz 2: Acute Angles and Right Triangles

An observer for a radar station is located at the origin of a coordinate system. For the point given, find the bearing of an airplane located at that point. Express the bearing using both methods. -(0, )

Find the sign of the following. - $\csc ( - \theta )$ , given that $\theta$ is in the interval $\left( 90 ^ { \circ } , 180 ^ { \circ } \right)$ .

Find the sign of the following. -cos (ϴ + 180°) , given that ϴ is in the interval (90°, 180°) .

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary. -cos 47° cos 133° - sin 47° sin 133°

Determine whether the statement is true or false. -sin 390° = sin 120° cos 270° + cos 120° sin 270°

Solve the right triangle. -A = 19° 17´, c = 287 ft, C = 90° Round side lengths to two decimal places, if necessary.

Solve the right triangle. -a = 3.6 m, B = 30.3°, C = 90° Round values to one decimal place.

Find the sign of the following. - $\tan \frac { \theta } { 2 }$ , given that $\theta$ is in the interval $\left( 90 ^ { \circ } , 180 ^ { \circ } \right)$ .

Find a solution for the equation. Assume that all angles are acute angles. -tan(3ϴ + 16°) = cot(ϴ + 4°)

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary. -cos 39° 5' cos 50° 55' - sin 39° 5' sin 50° 55'

Find a value of in [0°, 90°] that satisfies the statement. Leave answer in decimal degrees rounded to seven decimal places, if necessary. -csc ϴ = 1.7437345

Determine whether the statement is true or false. -cos 60° = cos 180° - cos 120°

Solve the problem. -From a balloon 1037 feet high, the angle of depression to the ranger headquarters is 58°41'. How far is the headquarters from a point on the ground directly below the balloon (to the nearest foot) ?