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Precalculus Trigonometry Test 2 – Step-by-Step Study Guidance

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Q1. Identify the amplitude, frequency, period, and shifts of the equation

Background

Topic: Graphs of Trigonometric Functions

This question tests your understanding of how to analyze the key features of a cosine function, including amplitude, period, frequency, phase shift, and vertical shift.

Key Terms and Formulas

  • Amplitude: in

  • Period:

  • Frequency: (number of cycles per units)

  • Phase Shift:

  • Vertical Shift:

Step-by-Step Guidance

  1. Compare the given equation to the standard form and identify , , , and .

  2. Find the amplitude by taking the absolute value of .

  3. Calculate the period using .

  4. Determine the phase shift using .

  5. Identify the vertical shift as .

Try solving on your own before revealing the answer!

Final Answer:

  • Amplitude: $5$

  • Period:

  • Frequency:

  • Phase Shift: (to the left)

  • Vertical Shift: $4$ (up)

We matched the equation to the standard form and used the formulas above to find each feature.

Q2. Graph one basic period of the equation

Background

Topic: Graphing Trigonometric Functions

This question asks you to sketch the graph of a transformed cosine function over one period, showing all key features.

Key Terms and Formulas

  • Basic period:

  • Key points: maximum, minimum, and midline crossings

  • Transformations: amplitude, phase shift, vertical shift

Step-by-Step Guidance

  1. Identify the amplitude, period, phase shift, and vertical shift (from Q1).

  2. Determine the starting and ending -values for one period, considering the phase shift.

  3. Mark the midline at and plot the maximum and minimum values using the amplitude.

  4. Divide the period into four equal intervals to find key points (maximum, minimum, intercepts).

  5. Sketch the curve, making sure to reflect the negative amplitude and all shifts.

Try solving on your own before revealing the answer!

Final Answer:

The graph is a cosine curve with amplitude $5, phase shift , and vertical shift $4y = 9y = -1\pix = -\pix = 3\pi$.

Q3. Graph one basic period of the equation

Background

Topic: Graphing Reciprocal Trigonometric Functions

This question tests your ability to graph the secant function, which is the reciprocal of cosine, including all transformations.

Key Terms and Formulas

  • Secant:

  • Vertical asymptotes occur where

  • Use the corresponding cosine graph as a guide

Step-by-Step Guidance

  1. Graph the related cosine function as a guide.

  2. Identify where the cosine function crosses its midline (these are the vertical asymptotes for secant).

  3. Plot the secant curves above the maxima and below the minima of the cosine graph.

  4. Mark the vertical asymptotes and sketch the basic period, showing the correct transformations.

Try solving on your own before revealing the answer!

Final Answer:

The secant graph has the same period, phase shift, and vertical shift as the cosine graph. Vertical asymptotes occur where the cosine function crosses its midline, and the secant curves open upward and downward from the maxima and minima, respectively.

Q4. Identify the amplitude, frequency, period, and shifts of the equation

Background

Topic: Graphs of Sine Functions

This question asks you to analyze the key features of a sine function with multiple transformations.

Key Terms and Formulas

  • Amplitude: in

  • Period:

  • Frequency:

  • Phase Shift:

  • Vertical Shift:

Step-by-Step Guidance

  1. Match the equation to the standard form and identify , , , and .

  2. Find the amplitude by taking .

  3. Calculate the period using .

  4. Determine the phase shift using .

  5. Identify the vertical shift as .

Try solving on your own before revealing the answer!

Final Answer:

  • Amplitude: $3$

  • Period:

  • Frequency: $2$

  • Phase Shift: (to the right)

  • Vertical Shift: (down)

Q5. Graph one basic period of the equation

Background

Topic: Graphing Sine Functions

This question asks you to sketch the graph of a transformed sine function over one period, showing all key features.

Key Terms and Formulas

  • Amplitude:

  • Period:

  • Phase Shift:

  • Vertical Shift:

Step-by-Step Guidance

  1. Identify the amplitude, period, phase shift, and vertical shift (from Q4).

  2. Determine the starting and ending -values for one period, considering the phase shift.

  3. Mark the midline at and plot the maximum and minimum values using the amplitude.

  4. Divide the period into four equal intervals to find key points (maximum, minimum, intercepts).

  5. Sketch the curve, making sure to reflect all transformations.

Try solving on your own before revealing the answer!

Final Answer:

The graph is a sine curve with amplitude $3\pi\frac{\pi}{4}-1y = 2y = -4x = \frac{\pi}{4}x = \frac{5\pi}{4}$.

Q6. Graph one basic period of the equation

Background

Topic: Graphing Reciprocal Trigonometric Functions

This question tests your ability to graph the cosecant function, which is the reciprocal of sine, including all transformations.

Key Terms and Formulas

  • Cosecant:

  • Vertical asymptotes occur where

  • Use the corresponding sine graph as a guide

Step-by-Step Guidance

  1. Graph the related sine function as a guide.

  2. Identify where the sine function crosses its midline (these are the vertical asymptotes for cosecant).

  3. Plot the cosecant curves above the maxima and below the minima of the sine graph.

  4. Mark the vertical asymptotes and sketch the basic period, showing the correct transformations.

Try solving on your own before revealing the answer!

Final Answer:

The cosecant graph has the same period, phase shift, and vertical shift as the sine graph. Vertical asymptotes occur where the sine function crosses its midline, and the cosecant curves open upward and downward from the maxima and minima, respectively.

Q7. Without a calculator, express as a function of its reference angle and find the exact value.

Background

Topic: Reference Angles and Exact Trigonometric Values

This question tests your ability to use reference angles and the unit circle to find exact trigonometric values.

Key Terms and Formulas

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

  • or

  • Quadrant IV: is positive

Step-by-Step Guidance

  1. Find the reference angle for .

  2. Determine the sign of based on the quadrant.

  3. Express in terms of the reference angle.

  4. Recall the exact values for and .

Try solving on your own before revealing the answer!

Final Answer:

Since is in the fourth quadrant, cotangent is positive, and the reference angle is .

Q8. Without a calculator, express as a function of its reference angle and find the exact value.

Background

Topic: Reference Angles and Exact Trigonometric Values

This question tests your ability to use reference angles and the unit circle to find exact sine values.

Key Terms and Formulas

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

  • Quadrant II: Sine is positive

  • Exact values for

Step-by-Step Guidance

  1. Find the reference angle for .

  2. Determine the sign of based on the quadrant.

  3. Express in terms of the reference angle.

  4. Recall the exact value for .

Try solving on your own before revealing the answer!

Final Answer:

Since is in the second quadrant, sine is positive, and the reference angle is .

Q9. The terminal side of an angle is given by the equation , . Determine the values of all six trigonometric functions of the angle.

Background

Topic: Trigonometric Functions of Angles Defined by Lines

This question tests your ability to find the six trigonometric functions for an angle whose terminal side lies on a given line, with a restriction on .

Key Terms and Formulas

  • Trigonometric functions: , , , , ,

  • Use a point on the line, and

  • Quadrant II:

Step-by-Step Guidance

  1. Choose a point on the line with (e.g., ).

  2. Find the corresponding value using the equation.

  3. Calculate .

  4. Write the six trigonometric functions in terms of , , and .

  5. Substitute the values and simplify each function.

Try solving on your own before revealing the answer!

Final Answer:

We used a point on the line and calculated .

Q10. Verify the identity:

Background

Topic: Trigonometric Identities

This question tests your ability to manipulate and verify trigonometric identities using algebraic and trigonometric properties.

Key Terms and Formulas

  • Pythagorean identity:

  • Factoring and combining like terms

Step-by-Step Guidance

  1. Start with the left side: .

  2. Factor from both terms.

  3. Use the Pythagorean identity to rewrite in terms of .

  4. Simplify the expression to see if it matches the right side.

Try solving on your own before revealing the answer!

Final Answer:

The identity is verified: .

Factoring and using the Pythagorean identity simplifies the left side to the right side.

Q11. Verify the identity:

Background

Topic: Trigonometric Identities

This question tests your ability to verify trigonometric identities by rewriting expressions in terms of sine and cosine.

Key Terms and Formulas

Step-by-Step Guidance

  1. Rewrite all terms in terms of sine and cosine.

  2. Distribute on the left side.

  3. Simplify each term and combine like terms if possible.

  4. Compare the result to the right side of the identity.

Try solving on your own before revealing the answer!

Final Answer:

The identity is verified: .

Expanding and simplifying both sides shows they are equal.

Q12. Find if and

Background

Topic: Trigonometric Functions and Quadrants

This question tests your ability to find one trigonometric function given another, using the Pythagorean identity and quadrant information.

Key Terms and Formulas

  • Pythagorean identity:

  • Quadrant IV: is negative, is positive

Step-by-Step Guidance

  1. Plug into the Pythagorean identity.

  2. Solve for .

  3. Take the square root to find .

  4. Determine the correct sign for based on the quadrant.

Try solving on your own before revealing the answer!

Final Answer:

Since is in the fourth quadrant, cosine is positive.

Q13. Find the exact value without a calculator:

Background

Topic: Exact Trigonometric Values and Angle Sum Formulas

This question tests your ability to use angle sum or difference identities to find exact trigonometric values.

Key Terms and Formulas

  • Angle sum identity:

  • Common angles: ,

Step-by-Step Guidance

  1. Express as a sum of two common angles (e.g., ).

  2. Apply the angle sum identity for cosine.

  3. Recall the exact values for , , , and .

  4. Substitute these values into the formula and simplify.

Try solving on your own before revealing the answer!

Final Answer:

We used the angle sum identity with and .

Q14. Find the exact value without a calculator:

Background

Topic: Exact Trigonometric Values and Angle Difference Formulas

This question tests your ability to use angle sum or difference identities to find exact trigonometric values.

Key Terms and Formulas

  • Angle difference identity:

  • Common angles: ,

Step-by-Step Guidance

  1. Express as a difference of two common angles (e.g., ).

  2. Apply the angle difference identity for sine.

  3. Recall the exact values for , , , and .

  4. Substitute these values into the formula and simplify.

Try solving on your own before revealing the answer!

Final Answer:

We used the angle difference identity with and .

Q15. Solve the right triangle if , . Round your answer to the appropriate value.

Background

Topic: Solving Right Triangles

This question tests your ability to use trigonometric ratios to solve for unknown sides and angles in a right triangle.

Key Terms and Formulas

  • Right triangle: one angle is

  • Law of sines and cosines, or basic trigonometric ratios

  • Sum of angles in a triangle:

Step-by-Step Guidance

  1. Label the triangle: , , (hypotenuse).

  2. Find angle using .

  3. Use to find the side opposite .

  4. Use to find the side adjacent to .

  5. Round your answers as instructed.

Try solving on your own before revealing the answer!

Final Answer:

  • Opposite side to :

  • Adjacent side to :

We used basic trigonometric ratios and rounded to three decimal places.

Q16. Solve the right triangle if , . Round your answer to the appropriate value.

Background

Topic: Solving Right Triangles

This question tests your ability to use trigonometric ratios and the Pythagorean theorem to solve for unknown sides and angles in a right triangle.

Key Terms and Formulas

  • Pythagorean theorem:

  • Trigonometric ratios: ,

  • Sum of angles in a triangle:

Step-by-Step Guidance

  1. Use the Pythagorean theorem to solve for the missing side .

  2. Find angle using .

  3. Find angle using .

  4. Round your answers as instructed.

Try solving on your own before revealing the answer!

Final Answer:

We used the Pythagorean theorem and trigonometric ratios, rounding to one decimal place.

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