Q1. Convert between degrees and radians

Background

Topic: Angle Measurement

This question tests your ability to convert angles from degrees to radians and vice versa, which is fundamental in trigonometry.

Key Terms and Formulas

• Degree: A unit for measuring angles, where a full circle is 360°.

• Radian: Another unit for measuring angles, where a full circle is radians.

Conversion formulas:

Step-by-Step Guidance

1. Identify whether you are converting from degrees to radians or radians to degrees.

2. Write down the appropriate conversion formula based on the direction of conversion.

3. Substitute the given value into the formula.

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

Try solving on your own before revealing the answer!

Q2. Determine the quadrant of an angle using the signs of sine and cosine

Background

Topic: Trigonometric Functions and the Unit Circle

This question tests your understanding of how the signs of sine and cosine relate to the four quadrants of the coordinate plane.

Key Terms and Concepts

• Quadrants: The four sections of the coordinate plane.

• Sine () and Cosine (): Their signs change depending on the quadrant.

Sign Chart:

• Quadrant I: ,

• Quadrant II: ,

• Quadrant III: ,

• Quadrant IV: ,

Step-by-Step Guidance

1. Identify the given signs for and .

2. Compare these signs to the sign chart for each quadrant.

3. Determine which quadrant matches the given signs.

Try solving on your own before revealing the answer!

Q3. Find the reference angle for a given angle

Background

Topic: Reference Angles

This question tests your ability to find the reference angle, which is the acute angle formed by the terminal side of the given angle and the x-axis.

Key Terms and Formulas

• Reference Angle: Always between (or $0$ and $\frac{\pi}{2}$ radians).

Formulas:

• Quadrant I:

• Quadrant II: or

• Quadrant III: or

• Quadrant IV: or

Step-by-Step Guidance

1. Determine the quadrant in which the angle lies.

2. Use the appropriate formula for that quadrant to find the reference angle.

3. Simplify the expression for the reference angle, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q4. Find all six trigonometric functions from a given terminal point

Background

Topic: Trigonometric Functions from Coordinates

This question tests your ability to find , , , , , and given a point on the terminal side of an angle.

Key Terms and Formulas

• Given point:

Step-by-Step Guidance

1. Identify the coordinates of the terminal point.

2. Calculate .

3. Set up the six trigonometric ratios using the formulas above.

4. Simplify each ratio as much as possible, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q5. Find the intercepts of trigonometric functions

Background

Topic: Graphs of Trigonometric Functions

This question tests your ability to find where a trigonometric function crosses the x-axis (x-intercepts) and y-axis (y-intercept).

Key Terms and Concepts

• x-intercept: Where

• y-intercept: Where

Step-by-Step Guidance

1. Set the trigonometric function equal to zero to find x-intercepts.

2. Solve for where the function equals zero.

3. For the y-intercept, substitute into the function and simplify.

Try solving on your own before revealing the answer!

Q6. Find the domain and range of a trigonometric function

Background

Topic: Domain and Range

This question tests your understanding of the possible input (domain) and output (range) values for trigonometric functions.

Key Terms and Concepts

• Domain: All possible values for which the function is defined.

• Range: All possible values the function can take.

Step-by-Step Guidance

1. Identify the trigonometric function (e.g., , , etc.).

2. Recall the standard domain and range for that function.

3. Consider any restrictions (e.g., division by zero for ).

Try solving on your own before revealing the answer!

Q7. Find the range of inverse trigonometric functions

Background

Topic: Inverse Trigonometric Functions

This question tests your knowledge of the output values (range) for , , and functions.

Key Terms and Ranges

• : Range is

• : Range is

• : Range is

Step-by-Step Guidance

1. Identify which inverse trigonometric function is being asked about.

2. Recall the standard range for that function.

3. Write the range in interval notation.

Try solving on your own before revealing the answer!

Q8. Find exact values of trigonometric expressions using identities and inverse properties

Background

Topic: Trigonometric Identities and Inverse Functions

This question tests your ability to use identities and inverse properties to find exact values, not decimal approximations.

Key Terms and Formulas

• Pythagorean identities:

• Reciprocal identities: , etc.

• Inverse properties:

Step-by-Step Guidance

1. Identify which identity or property applies to the expression.

2. Rewrite the expression using the appropriate identity.

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

Try solving on your own before revealing the answer!

Q9. Simplify trigonometric expressions using identities

Background

Topic: Trigonometric Identities

This question tests your ability to use identities to simplify complex trigonometric expressions.

Key Terms and Formulas

• Pythagorean, reciprocal, quotient, and cofunction identities.

Step-by-Step Guidance

1. Identify which identities can be applied to the given expression.

2. Rewrite the expression using these identities.

3. Simplify step by step, combining like terms or reducing fractions as needed.

Try solving on your own before revealing the answer!

Q10. Find the other five trigonometric values given one value and the quadrant

Background

Topic: Trigonometric Functions and Quadrants

This question tests your ability to use one trigonometric value and quadrant information to find the other five values.

Key Terms and Formulas

• Pythagorean identities

• Sign rules for each quadrant

Step-by-Step Guidance

1. Write the given value and identify the quadrant.

2. Use the Pythagorean identity to find another value (e.g., if is given, find ).

3. Determine the correct sign for each value based on the quadrant.

4. Find the reciprocal and quotient functions.

Try solving on your own before revealing the answer!

Q11. Use trigonometric ratios to express a function using a given value

Background

Topic: Trigonometric Ratios and Identities

This question tests your ability to express one trigonometric function in terms of another using identities and given values.

Key Terms and Formulas

• Quotient identities:

• Pythagorean identities

Step-by-Step Guidance

1. Write the given value (e.g., ).

2. Use the Pythagorean identity to find in terms of .

3. Express the desired function (e.g., ) in terms of .

Try solving on your own before revealing the answer!

Q12. Solve trigonometric equations

Background

Topic: Trigonometric Equations

This question tests your ability to solve equations involving trigonometric functions for all possible solutions.

Key Terms and Formulas

• Inverse trigonometric functions

• General solutions (e.g., )

Step-by-Step Guidance

1. Isolate the trigonometric function in the equation.

2. Apply the appropriate inverse function to both sides.

3. Write the general solution, considering the periodicity of the function.

Try solving on your own before revealing the answer!

Q13. Find an unknown side of a triangle using the Law of Sines, Law of Cosines, or trigonometric ratios

Background

Topic: Solving Triangles

This question tests your ability to use the Law of Sines, Law of Cosines, or basic trigonometric ratios to find missing sides in triangles.

Key Terms and Formulas

• Law of Sines:

• Law of Cosines:

• Basic ratios: , etc.

Step-by-Step Guidance

1. Identify which law or ratio applies based on the given information.

2. Write the appropriate formula and substitute the known values.

3. Rearrange the equation to solve for the unknown side.

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

Try solving on your own before revealing the answer!

Q14. Convert coordinates from polar to rectangular or vice versa

Background

Topic: Polar and Rectangular Coordinates

This question tests your ability to convert between polar and rectangular coordinates.

Key Terms and Formulas

• From polar to rectangular: ,

• From rectangular to polar: ,

Step-by-Step Guidance

1. Identify the given coordinates and the direction of conversion.

2. Write the appropriate formulas for conversion.

3. Substitute the given values into the formulas.

4. Simplify, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q15. Convert equations from polar to rectangular or vice versa

Background

Topic: Polar and Rectangular Equations

This question tests your ability to rewrite equations in polar form as rectangular, or vice versa, using coordinate relationships.

Key Terms and Formulas

Step-by-Step Guidance

1. Identify the given equation and the desired form.

2. Substitute the appropriate relationships (e.g., ) into the equation.

3. Simplify the equation to the desired form, but do not compute the final expression yet.

Try solving on your own before revealing the answer!

Q16. Find the position vector from a given pair of initial and terminal points

Background

Topic: Vectors

This question tests your ability to find the position vector from one point to another in the plane.

Key Terms and Formulas

• Initial point:

• Terminal point:

• Position vector:

Step-by-Step Guidance

1. Write the coordinates of the initial and terminal points.

2. Subtract the initial point from the terminal point for each component.

3. Write the resulting vector in component form.

Try solving on your own before revealing the answer!

Q17. Find a unit vector with the same direction as a given vector

Background

Topic: Vectors and Unit Vectors

This question tests your ability to find a vector of length 1 in the same direction as a given vector.

Key Terms and Formulas

• Given vector:

• Magnitude:

• Unit vector:

Step-by-Step Guidance

1. Write the components of the given vector.

2. Calculate the magnitude of the vector.

3. Divide each component by the magnitude to get the unit vector.

Try solving on your own before revealing the answer!

Q18. Find the magnitude of a vector

Background

Topic: Vector Magnitude

This question tests your ability to calculate the length of a vector given its components.

Key Terms and Formulas

• Given vector:

• Magnitude:

Step-by-Step Guidance

1. Write the components of the vector.

2. Square each component and add them together.

3. Take the square root of the sum to find the magnitude.

Try solving on your own before revealing the answer!

Q19. Solve story problems involving area of a segment, angle of elevation, or angle of depression

Background

Topic: Applications of Trigonometry

This question tests your ability to apply trigonometric concepts to real-world problems, such as finding areas or solving for unknowns using angles of elevation or depression.

Key Terms and Formulas

• Area of a sector: (with in radians)

• Angle of elevation/depression: Use right triangle trigonometry ()

Step-by-Step Guidance

1. Draw a diagram to represent the problem.

2. Label all known values and unknowns.

3. Write the appropriate formula for area or trigonometric ratio.

4. Substitute the known values into the formula and simplify, but do not compute the final value yet.

Try solving on your own before revealing the answer!