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Precalculus Test 1 Review – Step-by-Step Study Guidance

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Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

Q1. Arc Length: Find the missing quantity.

(a) Given: m, m, radians

Background

Topic: Arc Length of a Circle

This question tests your understanding of the relationship between arc length, radius, and central angle (in radians).

Key formula:

Where:

  • = arc length

  • = radius

  • = central angle (in radians)

Step-by-Step Guidance

  1. Write the formula for arc length: .

  2. Plug in the known values: .

  3. To solve for , divide both sides by $7$.

Try solving on your own before revealing the answer!

(b) Given: m, , radians

Background

Topic: Arc Length of a Circle

This part asks you to solve for the radius when given arc length and central angle.

Key formula:

Step-by-Step Guidance

  1. Start with the formula: .

  2. Plug in the known values: .

  3. To solve for , divide both sides by $7$.

Try solving on your own before revealing the answer!

Q2. Area of a Sector: Find the missing quantity.

(a) Given: sq ft, , radians

Background

Topic: Area of a Sector

This question tests your ability to use the formula for the area of a sector of a circle.

Key formula:

Step-by-Step Guidance

  1. Write the formula: .

  2. Plug in the known values: .

  3. Multiply both sides by $2$ to eliminate the fraction.

  4. Divide both sides by to solve for .

  5. Take the square root of both sides to solve for .

Try solving on your own before revealing the answer!

(b) Given: sq ft, ft,

Background

Topic: Area of a Sector (Angle in Degrees)

This part requires you to convert degrees to radians before using the area formula.

Key formula:

(with in radians)

Step-by-Step Guidance

  1. Convert to radians using .

  2. Plug the converted angle and into the area formula: .

  3. Simplify the expression to set up for calculation.

Try solving on your own before revealing the answer!

Q3. Angle Conversion

(a) Convert radians to degrees.

Background

Topic: Radian-Degree Conversion

This question tests your ability to convert between radians and degrees.

Key formula:

Step-by-Step Guidance

  1. Write the conversion formula: .

  2. Plug in for radians.

  3. Simplify the expression to set up for calculation.

Try solving on your own before revealing the answer!

(b) Convert to radians.

Background

Topic: Degree-Radian Conversion

Key formula:

Step-by-Step Guidance

  1. Write the conversion formula: .

  2. Plug in $84$ for degrees.

  3. Simplify the expression to set up for calculation.

Try solving on your own before revealing the answer!

Q4. Trigonometric Functions: Given , is acute. Find the remaining five trigonometric functions.

Background

Topic: Trigonometric Functions and Identities

This question tests your understanding of the relationships between the six trigonometric functions.

Key terms and relationships:

  • Other functions:

Step-by-Step Guidance

  1. Express in terms of : .

  2. Use to solve for .

  3. Find and using and .

  4. Find using .

Try solving on your own before revealing the answer!

Q5. The point is on the terminal side of an angle in standard position. Find the exact value of each of the six trig functions of .

Background

Topic: Trigonometric Functions from a Point

This question tests your ability to use the coordinates of a point to find all six trigonometric functions.

Key terms and formulas:

  • (distance from origin)

  • Reciprocal functions:

Step-by-Step Guidance

  1. Calculate .

  2. Find and .

  3. Find .

  4. Find as reciprocals of .

Try solving on your own before revealing the answer!

Q6. Find the exact value of the following:

(a)

Background

Topic: Evaluating Trigonometric Functions at Given Angles

This question tests your ability to evaluate trigonometric functions for angles outside the standard interval.

Key steps:

  • Find a coterminal angle between $0 by adding or subtracting multiples of .

  • Recall .

Step-by-Step Guidance

  1. Find a coterminal angle for by adding as needed.

  2. Evaluate at the coterminal angle.

  3. Take the reciprocal to find .

Try solving on your own before revealing the answer!

(b)

Background

Topic: Evaluating Trigonometric Functions

Key steps:

  • Find a coterminal angle between $0.

  • Evaluate at that angle.

Step-by-Step Guidance

  1. Subtract as needed to find a coterminal angle for .

  2. Evaluate at the coterminal angle.

Try solving on your own before revealing the answer!

Q7. Given and , find .

Background

Topic: Trigonometric Identities and Signs

This question tests your ability to use one trigonometric function to find another, considering the sign of the function.

Key relationships:

  • Use the Pythagorean identity:

Step-by-Step Guidance

  1. Let and for some (since ).

  2. Use the Pythagorean identity to solve for .

  3. Find and then .

Try solving on your own before revealing the answer!

Q8. Graph at least two cycles of the function .

Background

Topic: Graphing Trigonometric Functions

This question tests your ability to graph cosine functions with amplitude and vertical shift.

Key features:

  • Amplitude:

  • Vertical shift:

  • Period:

Step-by-Step Guidance

  1. Identify the amplitude, period, and vertical shift.

  2. Sketch the midline at .

  3. Mark the maximum and minimum values using amplitude and shift.

  4. Plot key points for two cycles and sketch the curve.

Try sketching the graph before checking the answer!

Q9. Graph at least two cycles of the function .

Background

Topic: Graphing Sine Functions with Transformations

Key features:

  • Amplitude: $1-1$)

  • Period:

  • Reflection over the x-axis due to the negative sign

Step-by-Step Guidance

  1. Calculate the period: .

  2. Note the amplitude and reflection.

  3. Mark key points for two cycles and sketch the curve.

Try sketching the graph before checking the answer!

Q10. Graph at least two cycles of the function .

Background

Topic: Graphing Cosine Functions with Phase Shift

Key features:

  • Amplitude: $1$

  • Period:

  • Phase shift: left by

Step-by-Step Guidance

  1. Identify the amplitude, period, and phase shift.

  2. Shift the basic cosine graph left by .

  3. Plot key points for two cycles and sketch the curve.

Try sketching the graph before checking the answer!

Q11. Graph at least two cycles of the function .

Background

Topic: Graphing Secant Functions

Key features:

  • Secant is the reciprocal of cosine.

  • Vertical shift:

  • Amplitude and period follow from the cosine base function.

Step-by-Step Guidance

  1. Graph as a reference.

  2. Draw vertical asymptotes where .

  3. Sketch the secant curves as the reciprocal of the cosine graph, shifted down by $8$.

Try sketching the graph before checking the answer!

Q12. Graph at least two cycles of the function .

Background

Topic: Graphing Cosecant Functions

Key features:

  • Cosecant is the reciprocal of sine.

  • Period: $6\frac{2\pi}{\frac{\pi}{3}}$)

  • Reflection over the x-axis due to the negative sign

Step-by-Step Guidance

  1. Graph as a reference.

  2. Draw vertical asymptotes where the sine function is zero.

  3. Sketch the cosecant curves as the reciprocal of the sine graph, reflected over the x-axis.

Try sketching the graph before checking the answer!

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