Step-by-Step Guidance for Plane Trigonometry Practice Test 1

Q2. Convert the following radian measure to degree measure: $\frac{7\pi}{4}$

Background

Topic: Radian-Degree Conversion

This question tests your ability to convert an angle from radians to degrees, a fundamental skill in trigonometry.

Key formula:

$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

Step-by-Step Guidance

1. Identify the radian value: $\frac{7\pi}{4}$.

2. Set up the conversion formula: $\text{Degrees} = \frac{7\pi}{4} \times \frac{180}{\pi}$.

3. Notice that $\pi$ in the numerator and denominator will cancel out.

Try solving on your own before revealing the answer!

Final Answer: 315°

$\frac{7\pi}{4} \times \frac{180}{\pi} = 315$

We multiplied the radian value by $\frac{180}{\pi}$ and simplified.

Q3. Identify the quadrant(s) for angles where $\sin \alpha > 0$ and $\cos \alpha < 0$.

Background

Topic: Signs of Trigonometric Functions in Quadrants

This question tests your understanding of which quadrants have positive or negative values for sine and cosine.

Key concepts:

• $\sin \alpha$ is positive in Quadrants I and II.

• $\cos \alpha$ is negative in Quadrants II and III.

Step-by-Step Guidance

1. Recall which quadrants have $\sin \alpha > 0$: Quadrants I and II.

2. Recall which quadrants have $\cos \alpha < 0$: Quadrants II and III.

3. Find the intersection: Which quadrant(s) satisfy both conditions?

Try solving on your own before revealing the answer!

Final Answer: 2

Only Quadrant II has $\sin \alpha > 0$ and $\cos \alpha < 0$.

Q4. Find the exact values of the six trigonometric functions for $-300^\circ$.

Background

Topic: Trigonometric Functions of Angles

This question tests your ability to find sine, cosine, tangent, cotangent, secant, and cosecant for a given angle, including negative angles.

Key formulas:

• $\sin(\theta)$, $\cos(\theta)$, $\tan(\theta)$, $\cot(\theta)$, $\sec(\theta)$, $\csc(\theta)$

• Use reference angles and the unit circle.

Step-by-Step Guidance

1. Find the reference angle for $-300^\circ$ by adding $360^\circ$ if needed: $-300^\circ + 360^\circ = 60^\circ$.

2. Determine the quadrant for $-300^\circ$ (or $60^\circ$ after adjustment).

3. Recall the values for $\sin(60^\circ)$, $\cos(60^\circ)$, $\tan(60^\circ)$, and their signs in the relevant quadrant.

4. Apply the sign rules for negative angles and calculate each function using exact values.

Try solving on your own before revealing the answer!

Final Answer:

$\sin(-300^\circ) = -\frac{\sqrt{3}}{2}$ $\cos(-300^\circ) = \frac{1}{2}$ $\tan(-300^\circ) = -\sqrt{3}$ $\cot(-300^\circ) = -\frac{1}{\sqrt{3}}$ $\csc(-300^\circ) = -\frac{2}{\sqrt{3}}$ $\sec(-300^\circ) = 2$

These are the exact values using reference angles and sign rules.

Q5. Find the exact value of each part labeled with a variable in the following figure: $b$, $f$, $c$, $g$.

Background

Topic: Trigonometric Values in Figures

This question tests your ability to use trigonometric relationships and possibly the Pythagorean theorem to find exact values in geometric figures.

Key formulas:

• Pythagorean theorem: $a^2 + b^2 = c^2$

• Trigonometric ratios: $\sin$, $\cos$, $\tan$

Step-by-Step Guidance

1. Identify the relationships between the variables in the figure (e.g., sides and angles).

2. Set up equations using trigonometric ratios or the Pythagorean theorem as appropriate.

3. Solve for each variable, rationalizing denominators if needed.

Try solving on your own before revealing the answer!

Final Answer:

Exact values for $b$, $f$, $c$, $g$ are found using trigonometric relationships and simplification.

Q6. Evaluate the expression: $2 \tan 120^\circ + 5 \sin 150^\circ - \cos 180^\circ$

Background

Topic: Evaluating Trigonometric Expressions

This question tests your ability to evaluate trigonometric functions for specific angles and combine them in an expression.

Key formulas:

• $\tan 120^\circ$, $\sin 150^\circ$, $\cos 180^\circ$

• Use exact values from the unit circle.

Step-by-Step Guidance

1. Find the exact value of $\tan 120^\circ$.

2. Find the exact value of $\sin 150^\circ$.

3. Find the exact value of $\cos 180^\circ$.

4. Plug these values into the expression and simplify, rationalizing denominators if needed.

Try solving on your own before revealing the answer!

Final Answer:

Exact value is found by substituting and simplifying the trigonometric values.

Q7. Suppose you are headed toward a plateau 20.2 meters high. If the angle of elevation to the top is $12.5^\circ$, how far are you from the base?

Background

Topic: Right Triangle Applications

This question tests your ability to use trigonometric ratios to solve real-world problems involving right triangles.

Key formula:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Step-by-Step Guidance

1. Identify the height of the plateau (opposite side): 20.2 meters.

2. Identify the angle of elevation: $12.5^\circ$.

3. Set up the equation: $\tan(12.5^\circ) = \frac{20.2}{x}$, where $x$ is the distance from the base.

4. Solve for $x$ by rearranging: $x = \frac{20.2}{\tan(12.5^\circ)}$.

Try solving on your own before revealing the answer!

Final Answer: 91.1 meters

We used the tangent ratio and solved for the adjacent side.

Q8. Find the distance from City A to City C given bearings and travel times.

Background

Topic: Law of Cosines and Bearings

This question tests your ability to use bearings and the Law of Cosines to find distances in navigation problems.

Key formula:

$c^2 = a^2 + b^2 - 2ab\cos(C)$

Step-by-Step Guidance

1. Calculate the distances from City A to B and B to C using speed and time.

2. Draw a diagram to visualize the bearings and angles.

3. Use the Law of Cosines to find the distance from City A to City C.

Try solving on your own before revealing the answer!

Final Answer: 132 miles

We calculated the sides and used the Law of Cosines.

Q9. Find the measure (in radians) of a central angle of a sector of area 14 square inches in a circle of radius 3.6 inches.

Background

Topic: Area of a Sector

This question tests your ability to relate the area of a sector to its central angle in radians.

Key formula:

$A = \frac{1}{2} r^2 \theta$

Step-by-Step Guidance

1. Identify the area $A = 14$ and radius $r = 3.6$.

2. Set up the formula: $14 = \frac{1}{2} (3.6)^2 \theta$.

3. Solve for $\theta$ by rearranging: $\theta = \frac{2 \times 14}{(3.6)^2}$.

Try solving on your own before revealing the answer!

Final Answer: 2.2 radians

We used the sector area formula and solved for the central angle.

Q10. The tires of a bicycle have radius 13.0 in. and are turning at the rate of 225 revolutions per min. How fast is the bicycle traveling in miles per hour?

Background

Topic: Linear Speed and Circular Motion

This question tests your ability to relate angular speed to linear speed and convert units.

Key formulas:

• Distance per revolution: $2\pi r$

• Linear speed: $v = \text{revolutions per min} \times \text{distance per revolution}$

• Unit conversions: inches to feet, feet to miles

Step-by-Step Guidance

1. Calculate the distance traveled in one revolution: $2\pi \times 13.0$ inches.

2. Multiply by 225 revolutions per minute to get inches per minute.

3. Convert inches per minute to miles per hour using unit conversions.

Try solving on your own before revealing the answer!

Final Answer: 17.4 mph

We calculated the linear speed and converted units appropriately.

Q11. Suppose point P is on a circle with radius $r = 8$ in., angular speed $\omega = \frac{\pi}{5}$ rad/min, and time $t = 10$ min. Complete parts (a)-(c).

Background

Topic: Angular and Linear Motion

This question tests your ability to relate angular speed, radius, and time to angle, arc length, and linear speed.

Key formulas:

• Angle generated: $\theta = \omega t$

• Distance traveled: $s = r \theta$

• Linear speed: $v = r \omega$

Step-by-Step Guidance

1. For (a): Calculate the angle generated: $\theta = \omega t = \frac{\pi}{5} \times 10$.

2. For (b): Calculate the distance traveled: $s = r \theta$.

3. For (c): Calculate the linear speed: $v = r \omega$.

Try solving on your own before revealing the answer!

Final Answer:

(a) $2\pi$ radians (b) $16\pi$ inches (c) $\frac{8\pi}{5}$ inches per minute

We used the angular and linear motion formulas for each part.