Q1. Two sides of a right triangle ABC (C is the right angle) are given: a = 2 and b = 3. Find sin(A). Give the exact answer with a rational denominator.

Background

Topic: Right Triangle Trigonometry

This question tests your understanding of how to use the definitions of trigonometric ratios in a right triangle, specifically the sine function.

Key Terms and Formulas:

• Opposite side: The side opposite the angle in question (here, angle A).

• Hypotenuse: The longest side of the right triangle, opposite the right angle.

• $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}$

• Pythagorean Theorem: $a^2 + b^2 = c^2$ (where c is the hypotenuse)

Step-by-Step Guidance

1. Identify which sides correspond to a, b, and c. Typically, in triangle ABC with right angle at C, sides a and b are the legs, and c is the hypotenuse.

2. Use the Pythagorean Theorem to find the length of the hypotenuse: $c = \sqrt{a^2 + b^2}$.

3. Determine which side is opposite angle A (it will be side b if A is at vertex A).

4. Set up the sine ratio: $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}$.

5. Plug in the values for the opposite side and the hypotenuse, but do not simplify to the final answer yet.

Try solving on your own before revealing the answer!

Q2. Convert the angle (given in degrees, minutes, and seconds) to a decimal in degrees. Round your answer to two decimal places.

Background

Topic: Angle Measurement Conversion

This question tests your ability to convert an angle from degrees, minutes, and seconds (DMS) to decimal degrees.

Key Terms and Formulas:

• Degrees (°), Minutes ('), Seconds (")

• 1 degree = 60 minutes, 1 minute = 60 seconds

• Decimal degrees = degrees + (minutes/60) + (seconds/3600)

Step-by-Step Guidance

1. Identify the values for degrees, minutes, and seconds from the given angle.

2. Divide the number of minutes by 60 to convert to degrees.

3. Divide the number of seconds by 3600 to convert to degrees.

4. Add all three values together to get the decimal degree measure.

5. Set up the calculation, but do not compute the final decimal value yet.

Try solving on your own before revealing the answer!

Q3. Convert the angle (given in decimal degrees) to degrees, minutes, and seconds (DMS) form. Round your answer to the nearest second.

Background

Topic: Angle Measurement Conversion

This question tests your ability to convert a decimal degree value into degrees, minutes, and seconds.

Key Terms and Formulas:

• Degrees (°), Minutes ('), Seconds (")

• 1 degree = 60 minutes, 1 minute = 60 seconds

Step-by-Step Guidance

1. Take the whole number part as the degrees.

2. Multiply the decimal part by 60 to get the minutes.

3. Take the decimal part of the minutes and multiply by 60 to get the seconds.

4. Round the seconds to the nearest whole number.

5. Set up the conversion, but do not write the final DMS value yet.

Try solving on your own before revealing the answer!

Q4. Convert the angle (given in degrees) to radians.

Background

Topic: Angle Measurement Conversion

This question tests your ability to convert an angle from degrees to radians.

Key Terms and Formulas:

• Degrees (°), Radians (rad)

• $1\text{ radian} = \frac{180}{\pi}$ degrees

• To convert degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

Step-by-Step Guidance

1. Write down the given angle in degrees.

2. Set up the conversion formula: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$.

3. Plug in the degree value into the formula.

4. Simplify the expression as much as possible, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q5. Convert the angle (given in radians) to degrees.

Background

Topic: Angle Measurement Conversion

This question tests your ability to convert an angle from radians to degrees.

Key Terms and Formulas:

• Degrees (°), Radians (rad)

• $1\text{ radian} = \frac{180}{\pi}$ degrees

• To convert radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Step-by-Step Guidance

1. Write down the given angle in radians.

2. Set up the conversion formula: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$.

3. Plug in the radian value into the formula.

4. Simplify the expression as much as possible, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q6. Given that $\sin(\theta) = a$ and $\cos(\theta) = b$, find $\tan(\theta)$.

Background

Topic: Fundamental Trigonometric Identities

This question tests your understanding of the relationship between sine, cosine, and tangent.

Key Terms and Formulas:

• $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Step-by-Step Guidance

1. Recall the definition of tangent in terms of sine and cosine.

2. Set up the formula: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

3. Substitute the given values for $\sin(\theta)$ and $\cos(\theta)$.

4. Simplify the expression, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q7. Use fundamental identities and/or the complementary angle theorem to find the exact value of $\sin(90^\circ - x)$.

Background

Topic: Cofunction Identities

This question tests your knowledge of the cofunction identities for sine and cosine.

Key Terms and Formulas:

• Cofunction identity: $\sin(90^\circ - x) = \cos(x)$

Step-by-Step Guidance

1. Recall the cofunction identity for sine and cosine.

2. Apply the identity to rewrite $\sin(90^\circ - x)$ in terms of cosine.

3. Express the answer in terms of $\cos(x)$, but do not substitute any values yet.

Try solving on your own before revealing the answer!

Q8. If $\theta = 45^\circ$, find the exact value of $\sin(2\theta)$ when $\sin(\theta) = \cos(\theta)$. Do not use a calculator.

Background

Topic: Double Angle Identities

This question tests your ability to use the double angle identity for sine and recognize special angle values.

Key Terms and Formulas:

• Double angle identity: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

• For $\theta = 45^\circ$, $\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$

Step-by-Step Guidance

1. Write the double angle identity for sine: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.

2. Substitute $\theta = 45^\circ$ and $\sin(\theta) = \cos(\theta)$ into the formula.

3. Plug in the value for $\sin(45^\circ)$ and $\cos(45^\circ)$, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q9. Use a calculator to find the approximate value of $\sin(30^\circ)$. Round your answer to two decimal places.

Background

Topic: Evaluating Trigonometric Functions

This question tests your ability to use a calculator to find the value of a trigonometric function and round appropriately.

Key Terms and Formulas:

• $\sin(\theta)$: Sine function

• Calculator in degree mode

Step-by-Step Guidance

1. Ensure your calculator is set to degree mode.

2. Enter $30$ and press the sine function key.

3. Read the value from the calculator and round to two decimal places, but do not write the final value yet.

Try solving on your own before revealing the answer!

Q10. The point (3, 4) is on the terminal side of the angle $\theta$. Find the exact value of $\sin(\theta)$.

Background

Topic: Trigonometric Functions and the Unit Circle

This question tests your ability to find trigonometric function values given a point on the terminal side of an angle.

Key Terms and Formulas:

• For a point $(x, y)$ on the terminal side: $\sin(\theta) = \frac{y}{r}$, where $r = \sqrt{x^2 + y^2}$

Step-by-Step Guidance

1. Identify $x = 3$ and $y = 4$.

2. Calculate $r = \sqrt{x^2 + y^2}$.

3. Set up $\sin(\theta) = \frac{y}{r}$.

4. Plug in the values for $y$ and $r$, but do not simplify to the final answer yet.

Try solving on your own before revealing the answer!

Q11. A twenty-five foot ladder just reaches the top of a house and forms an angle of $60^\circ$ with the wall of the house. How tall is the house? Round your answer to the nearest 0.1 foot.

Background

Topic: Right Triangle Applications

This question tests your ability to apply trigonometric ratios to solve real-world right triangle problems.

Key Terms and Formulas:

• Opposite side: Height of the house

• Hypotenuse: Length of the ladder

• $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Step-by-Step Guidance

1. Let $h$ be the height of the house (opposite side).

2. Set up the equation: $\sin(60^\circ) = \frac{h}{25}$.

3. Solve for $h$: $h = 25 \times \sin(60^\circ)$.

4. Plug in the value for $\sin(60^\circ)$, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q12. Name the quadrant in which the angle $\theta$ lies if $\sin(\theta) < 0$ and $\cos(\theta) > 0$.

Background

Topic: Signs of Trigonometric Functions by Quadrant

This question tests your knowledge of the signs of sine and cosine in the four quadrants.

Key Terms and Formulas:

• Quadrant I: $\sin > 0$, $\cos > 0$

• Quadrant II: $\sin > 0$, $\cos < 0$

• Quadrant III: $\sin < 0$, $\cos < 0$

• Quadrant IV: $\sin < 0$, $\cos > 0$

Step-by-Step Guidance

1. Recall the signs of sine and cosine in each quadrant.

2. Identify which quadrant has $\sin(\theta) < 0$ and $\cos(\theta) > 0$.

3. State the quadrant, but do not write the final answer yet.

Try solving on your own before revealing the answer!

Q13. Determine the sign of $\sin(-120^\circ)$.

Background

Topic: Signs of Trigonometric Functions

This question tests your understanding of the sign of the sine function for negative angles.

Key Terms and Formulas:

• Sine is negative in Quadrants III and IV.

• $\sin(-\theta) = -\sin(\theta)$ (odd function property)

Step-by-Step Guidance

1. Recognize that $-120^\circ$ is a negative angle measured clockwise from the positive x-axis.

2. Use the odd property: $\sin(-120^\circ) = -\sin(120^\circ)$.

3. Determine the sign of $\sin(120^\circ)$ (which is positive), so $\sin(-120^\circ)$ will be negative.

4. State the sign, but do not write the final answer yet.

Try solving on your own before revealing the answer!

Q14. Find the reference angle of $-135^\circ$.

Background

Topic: Reference Angles

This question tests your ability to find the reference angle for a given angle, especially when the angle is negative.

Key Terms and Formulas:

• Reference angle: The acute angle formed by the terminal side of the given angle and the x-axis.

• For negative angles, add $360^\circ$ to find the coterminal positive angle.

• Reference angle formula depends on the quadrant.

Step-by-Step Guidance

1. Add $360^\circ$ to $-135^\circ$ to find the coterminal angle.

2. Determine which quadrant the angle lies in.

3. Use the appropriate formula to find the reference angle (e.g., $\text{Reference angle} = 180^\circ - \theta$ for Quadrant II).

4. Set up the calculation, but do not write the final value yet.

Try solving on your own before revealing the answer!

Q15. Use the reference angle to find the exact value of $\sin(-225^\circ)$.

Background

Topic: Reference Angles and Trigonometric Values

This question tests your ability to use reference angles and the signs of trigonometric functions to find exact values.

Key Terms and Formulas:

• Reference angle: The acute angle formed with the x-axis.

• Sign of sine in the quadrant where $-225^\circ$ lies.

• Exact values for $\sin(45^\circ)$ and $\sin(225^\circ)$.

Step-by-Step Guidance

1. Find the reference angle for $-225^\circ$ (add $360^\circ$ to get a positive coterminal angle).

2. Determine the sign of sine in the quadrant where the angle lies.

3. Use the reference angle to find the value, but do not write the final answer yet.

Try solving on your own before revealing the answer!

Q16. Find the exact value of $\cos(\theta)$ given that $\sin(\theta) = -\frac{3}{5}$ and $\pi < \theta < \frac{3\pi}{2}$.

Background

Topic: Trigonometric Values and Quadrants

This question tests your ability to use the Pythagorean identity and knowledge of quadrants to find the cosine value.

Key Terms and Formulas:

• Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$

• Quadrant III: Both sine and cosine are negative

Step-by-Step Guidance

1. Plug the given value of $\sin(\theta)$ into the Pythagorean identity.

2. Solve for $\cos^2(\theta)$.

3. Take the square root to find $\cos(\theta)$, remembering to choose the correct sign based on the quadrant.

4. Set up the expression, but do not write the final value yet.

Try solving on your own before revealing the answer!

Q17. If $\tan(\theta) = 1$, find $\sin(\theta + \pi)$.

Background

Topic: Trigonometric Identities and Periodicity

This question tests your understanding of the periodic properties of sine and the tangent function.

Key Terms and Formulas:

• $\sin(\theta + \pi) = -\sin(\theta)$

• If $\tan(\theta) = 1$, $\theta = 45^\circ$ or $225^\circ$ (or $\frac{\pi}{4}$, $\frac{5\pi}{4}$)

Step-by-Step Guidance

1. Recall the periodic property: $\sin(\theta + \pi) = -\sin(\theta)$.

2. Find $\sin(\theta)$ for $\tan(\theta) = 1$ (i.e., $\theta = 45^\circ$ or $\frac{\pi}{4}$).

3. Apply the property to find $\sin(\theta + \pi)$, but do not write the final value yet.

Try solving on your own before revealing the answer!

Q18. Use the even-odd properties to find the exact values of $\cos(-x)$ and $\sin(-x)$.

Background

Topic: Even-Odd Properties of Trigonometric Functions

This question tests your understanding of the symmetry properties of sine and cosine.

Key Terms and Formulas:

• Even function: $f(-x) = f(x)$ (cosine is even)

• Odd function: $f(-x) = -f(x)$ (sine is odd)

Step-by-Step Guidance

1. Recall that cosine is an even function: $\cos(-x) = \cos(x)$.

2. Recall that sine is an odd function: $\sin(-x) = -\sin(x)$.

3. Write the expressions for each, but do not substitute any values yet.

Try solving on your own before revealing the answer!

Q19. Find the exact value of $\cos(2\pi - x)$.

Background

Topic: Trigonometric Identities and Periodicity

This question tests your understanding of the periodic properties of the cosine function.

Key Terms and Formulas:

• $\cos(2\pi - x) = \cos(x)$ (cosine is periodic with period $2\pi$ and is an even function)

Step-by-Step Guidance

1. Recall the periodic property: $\cos(2\pi - x) = \cos(x)$.

2. Write the expression using the property, but do not substitute any values yet.

Try solving on your own before revealing the answer!

Q20. Graph $y = \sin(x - \frac{\pi}{2})$.

Background

Topic: Graphs of Trigonometric Functions

This question tests your ability to graph a sine function with a phase shift.

Key Terms and Formulas:

• Standard sine graph: $y = \sin(x)$

• Phase shift: $x - c$ shifts the graph right by $c$ units

Step-by-Step Guidance

1. Identify the phase shift: $x - \frac{\pi}{2}$ means a shift to the right by $\frac{\pi}{2}$.

2. Sketch the basic sine curve.

3. Shift every point on the sine curve to the right by $\frac{\pi}{2}$ units.

4. Label key points (e.g., maximum, minimum, intercepts), but do not draw the final graph yet.

Try sketching the graph on your own before revealing the answer!

Q21. Find the amplitude, period, and phase shift of $y = -2\sin(x + \frac{\pi}{4})$.

Background

Topic: Graphs of Trigonometric Functions

This question tests your ability to identify amplitude, period, and phase shift from the equation of a sine function.

Key Terms and Formulas:

• General form: $y = a\sin(bx + c)$

• Amplitude: $|a|$

• Period: $\frac{2\pi}{|b|}$

• Phase shift: $-\frac{c}{b}$

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ from the equation.

2. Calculate the amplitude as $|a|$.

3. Calculate the period as $\frac{2\pi}{|b|}$.

4. Calculate the phase shift as $-\frac{c}{b}$, but do not write the final values yet.

Try solving on your own before revealing the answer!