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Precalculus Exam 1 Review – Step-by-Step Guidance

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Q1(a). If an angle in standard position is between and , in which quadrant is the angle?

Background

Topic: Angles in Standard Position and Quadrants

This question tests your understanding of how to determine the quadrant of an angle given in radians, especially when the angle is greater than (i.e., more than one full rotation).

Key Terms and Formulas:

  • Standard Position: An angle whose vertex is at the origin and whose initial side lies along the positive x-axis.

  • Quadrants: The four sections of the coordinate plane, labeled I, II, III, IV, counterclockwise from the positive x-axis.

  • One full rotation: radians.

Step-by-Step Guidance

  1. First, recognize that and are both greater than , so the angle has completed at least one full rotation.

  2. Find the coterminal angle between $0 by subtracting multiples of $2\pi$ from the given bounds.

  3. Determine which quadrant the resulting angle lies in by comparing it to $0\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}.

Try solving on your own before revealing the answer!

Q1(b). In which quadrant is angle (in standard position)?

Background

Topic: Angles and Quadrants (Degrees)

This question tests your ability to determine the quadrant of an angle given in degrees.

Key Terms and Formulas:

  • Quadrants: I: to , II: $90^\circ$ to , III: $180^\circ$ to , IV: $270^\circ$ to .

Step-by-Step Guidance

  1. Compare to the quadrant boundaries: , , , , .

  2. Identify which interval falls into to determine the quadrant.

Try solving on your own before revealing the answer!

Q2. Are the angles and coterminal angles? (Circle True or False)

Background

Topic: Coterminal Angles

This question tests your understanding of coterminal angles, which are angles that share the same terminal side.

Key Terms and Formulas:

  • Coterminal Angles: Two angles are coterminal if their difference is a multiple of .

  • General formula: , where is an integer.

Step-by-Step Guidance

  1. Subtract from to find their difference.

  2. Check if the difference is a multiple of .

Try solving on your own before revealing the answer!

Q3. Find the least positive angle coterminal with .

Background

Topic: Coterminal Angles (Degrees)

This question tests your ability to find coterminal angles by reducing a given angle to its equivalent between and .

Key Terms and Formulas:

  • Coterminal Angle: , where is the number of full rotations.

Step-by-Step Guidance

  1. Divide by to determine how many full rotations are in the angle.

  2. Subtract as many times as needed to get an angle between and $360^\circ$.

Try solving on your own before revealing the answer!

Q4. Find the arc length if a circle of radius 20 inches subtends a central angle of . Give an exact value, not a decimal approximation.

Background

Topic: Arc Length (Radians)

This question tests your ability to use the arc length formula for a circle when the central angle is given in radians.

Key Terms and Formulas:

  • Arc Length Formula: , where is arc length, is radius, and is the central angle in radians.

Step-by-Step Guidance

  1. Identify the radius inches and the angle radians.

  2. Plug these values into the formula .

  3. Multiply $20\frac{4\pi}{5}$ to set up the calculation for arc length.

Try solving on your own before revealing the answer!

Q5. If a wheel rolls a distance of 50 inches as it turns through an angle of , what is the radius of the wheel? Round your answer to two decimal places.

Background

Topic: Arc Length and Radius (Degrees)

This question tests your ability to relate arc length, radius, and central angle (in degrees) for a circle.

Key Terms and Formulas:

  • Arc Length Formula (Radians):

  • Convert degrees to radians:

Step-by-Step Guidance

  1. Convert to radians using .

  2. Set up the arc length formula with inches and in radians.

  3. Solve for by rearranging the formula: .

Try solving on your own before revealing the answer!

Q6. Convert into degrees.

Background

Topic: Radian-Degree Conversion

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

Key Terms and Formulas:

  • Conversion Formula:

Step-by-Step Guidance

  1. Multiply by .

  2. Simplify the expression to find the degree measure.

Try solving on your own before revealing the answer!

Q7. If the point is on the terminal side of an angle on the unit circle in standard position, find the exact values of all six trigonometric functions for $\theta$. Rationalize all denominators as necessary.

Background

Topic: Trigonometric Functions from a Point

This question tests your ability to use the coordinates of a point to find the six trigonometric functions for an angle in standard position.

Key Terms and Formulas:

  • Given a point ,

  • , ,

  • , ,

Step-by-Step Guidance

  1. Calculate .

  2. Find , , and using the definitions above.

  3. Find , , and as reciprocals of the basic functions.

  4. Rationalize denominators as needed.

Try solving on your own before revealing the answer!

Q8. Find the exact value of the following trigonometric functions:

  • (a)

  • (b)

  • (c)

  • (d)

  • (e)

  • (f)

Background

Topic: Evaluating Trigonometric Functions at Special Angles

This question tests your knowledge of the unit circle and special angle values for sine, cosine, cotangent, and secant.

Key Terms and Formulas:

  • Unit circle values for $0, , , , , , (and their radian equivalents).

  • ,

Step-by-Step Guidance

  1. For each part, locate the angle on the unit circle and recall the corresponding sine, cosine, cotangent, or secant value.

  2. For negative angles, use even/odd properties as needed.

  3. Express all answers in exact form (using radicals or integers as appropriate).

Try solving on your own before revealing the answer!

Q9. Use your calculator to estimate to two decimal places.

Background

Topic: Calculator Evaluation of Trigonometric Functions

This question tests your ability to use a calculator to find the value of a trigonometric function (cosecant) for a given angle in degrees.

Key Terms and Formulas:

Step-by-Step Guidance

  1. Find using your calculator (make sure it is in degree mode).

  2. Take the reciprocal of the result to find .

  3. Round your answer to two decimal places.

Try solving on your own before revealing the answer!

Q10. Give the domain and range of the cosine function, in interval notation.

Background

Topic: Domain and Range of Trigonometric Functions

This question tests your understanding of the set of all possible input (domain) and output (range) values for the cosine function.

Key Terms and Formulas:

  • Domain: All real numbers for .

  • Range: for .

Step-by-Step Guidance

  1. Recall that cosine is defined for all real numbers.

  2. Recall that the maximum and minimum values of cosine are $1-1$, respectively.

  3. Express the domain and range in interval notation.

Try solving on your own before revealing the answer!

Q11(a). Find the quadrant in which the angle lies if .

Background

Topic: Signs of Trigonometric Functions in Quadrants

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

Key Terms and Formulas:

  • Quadrant I: ,

  • Quadrant II: ,

  • Quadrant III: ,

  • Quadrant IV: ,

Step-by-Step Guidance

  1. Identify which quadrant(s) have .

  2. Within those, determine where .

Try solving on your own before revealing the answer!

Q11(b). Find the quadrant in which the angle lies if .

Background

Topic: Signs of Trigonometric Functions in Quadrants

This question tests your ability to use the signs of tangent and cosecant to determine the quadrant.

Key Terms and Formulas:

Step-by-Step Guidance

  1. Determine in which quadrants (where sine and cosine have the same sign).

  2. Within those, find where (i.e., ).

Try solving on your own before revealing the answer!

Q12. If the point is on the terminal side of an angle in standard position, find the exact values of and . Rationalize all denominators as necessary.

Background

Topic: Trigonometric Functions from a Point

This question tests your ability to use the coordinates of a point to find sine and tangent for an angle in standard position.

Key Terms and Formulas:

  • Given a point ,

  • ,

Step-by-Step Guidance

  1. Calculate .

  2. Find and using the definitions above.

  3. Rationalize denominators as needed.

Try solving on your own before revealing the answer!

Q13. Which of the following is always true?

  • (i)

  • (ii)

  • (iii)

Background

Topic: Pythagorean Trigonometric Identities

This question tests your knowledge of fundamental trigonometric identities involving tangent and secant.

Key Terms and Formulas:

  • Pythagorean Identity:

Step-by-Step Guidance

  1. Recall the Pythagorean identities for sine, cosine, tangent, and secant.

  2. Compare each option to the correct identity.

Try solving on your own before revealing the answer!

Q14. If and , find and .

Background

Topic: Trigonometric Functions Given One Value and a Sign Condition

This question tests your ability to use the value of one trigonometric function and a sign condition to find other functions.

Key Terms and Formulas:

  • ,

  • Pythagorean Identity:

Step-by-Step Guidance

  1. Let , so , .

  2. Use the Pythagorean identity to solve for and thus .

  3. Use the sign condition () to determine the correct sign for .

  4. Find as the reciprocal of .

Try solving on your own before revealing the answer!

Q15(a). If , what is ?

Background

Topic: Even/Odd Properties of Trigonometric Functions

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

Key Terms and Formulas:

  • Sine is an odd function:

Step-by-Step Guidance

  1. Recall the property of odd functions for sine.

  2. Apply this property to .

Try solving on your own before revealing the answer!

Q15(b). If , what is ?

Background

Topic: Even/Odd Properties of Trigonometric Functions

This question tests your understanding of the symmetry properties of the secant function.

Key Terms and Formulas:

  • Secant is an even function:

Step-by-Step Guidance

  1. Recall the property of even functions for secant.

  2. Apply this property to .

Try solving on your own before revealing the answer!

Q16. Graph one period of . Label five points on your graph. State the amplitude and period.

Background

Topic: Graphing Cosine Functions

This question tests your ability to graph a transformed cosine function and identify amplitude and period.

Key Terms and Formulas:

  • General form:

  • Amplitude:

  • Period:

Step-by-Step Guidance

  1. Identify , , .

  2. Find the amplitude and period using the formulas above.

  3. Determine five key points by evaluating the function at , , , , .

Try solving on your own before revealing the answer!

Q17. Graph one period of . Label five points on your graph. State the amplitude and period.

Background

Topic: Graphing Sine Functions

This question tests your ability to graph a transformed sine function and identify amplitude and period.

Key Terms and Formulas:

  • General form:

  • Amplitude:

  • Period:

Step-by-Step Guidance

  1. Identify , , .

  2. Find the amplitude and period using the formulas above.

  3. Determine five key points by evaluating the function at , , , , .

Try solving on your own before revealing the answer!

Q18. Graph one period of . Label five points. State the amplitude, period, and phase shift.

Background

Topic: Graphing Cosine Functions with Transformations

This question tests your ability to graph a cosine function with amplitude, phase shift, and vertical shift.

Key Terms and Formulas:

  • General form:

  • Amplitude:

  • Period:

  • Phase shift:

Step-by-Step Guidance

  1. Identify , , , .

  2. Find the amplitude, period, and phase shift using the formulas above.

  3. Determine five key points by evaluating the function at appropriate values.

Try solving on your own before revealing the answer!

Q19. Graph one period of . Label five points on your graph. State the amplitude, period, and phase shift.

Background

Topic: Graphing Sine Functions with Transformations

This question tests your ability to graph a sine function with amplitude, period, and phase shift.

Key Terms and Formulas:

  • General form:

  • Amplitude:

  • Period:

  • Phase shift:

Step-by-Step Guidance

  1. Identify , , , .

  2. Find the amplitude, period, and phase shift using the formulas above.

  3. Determine five key points by evaluating the function at appropriate values.

Try solving on your own before revealing the answer!

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