Q1(a). What is the domain of ?

Background

Topic: Trigonometric Functions – Domain

This question tests your understanding of the domain of the cosecant function, which is the reciprocal of the sine function.

Key Terms and Formulas

• The domain of a function is the set of all real numbers for which the function is defined.

Step-by-Step Guidance

1. Recall that is undefined wherever .

2. Determine for which values of the sine function equals zero.

3. Express these values in terms of , where is an integer.

4. State the domain as all real numbers except these excluded values.

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Q1(b). What is the period of the secant function?

Background

Topic: Trigonometric Functions – Periodicity

This question is about the period of the secant function, which is the reciprocal of the cosine function.

Key Terms and Formulas

• The period of is .

• The period of is the same as the period of .

Step-by-Step Guidance

1. Recall the period of the cosine function.

2. Understand that the secant function repeats its values over the same interval as the cosine function.

3. State the period for based on this information.

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Q1(c). What is the domain of ?

Background

Topic: Trigonometric Functions – Domain

This question is similar to 1(a), focusing on the reciprocal of the sine function.

Key Terms and Formulas

• is undefined wherever .

• Find all where .

Step-by-Step Guidance

1. Identify the values of where (these are the points to exclude from the domain).

2. Express these values in terms of .

3. State the domain as all real numbers except these points.

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Q1(d). List all y-intercepts of .

Background

Topic: Trigonometric Functions – Intercepts

This question asks you to find where the cotangent function crosses the y-axis.

Key Terms and Formulas

• The y-intercept occurs where .

Step-by-Step Guidance

1. Set and evaluate .

2. Recall that , so is undefined.

3. Consider if there are any other values where the graph crosses the y-axis.

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Q2. Graph one period of . Label five points. State the amplitude, period, and phase shift.

Background

Topic: Graphing Trigonometric Functions

This question tests your ability to analyze and graph a transformed cosine function, identifying amplitude, period, and phase shift.

Key Terms and Formulas

• General form:

• Amplitude:

• Period:

• Phase shift:

Step-by-Step Guidance

1. Identify , , and from the equation .

2. Calculate the amplitude as .

3. Find the period using .

4. Determine the phase shift using .

5. Choose five key -values within one period and compute the corresponding -values for labeling points on the graph.

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Q3. Graph one period of . Label five points. State the amplitude, period, and phase shift.

Background

Topic: Graphing Trigonometric Functions

This question is about graphing a sine function with amplitude and period changes.

Key Terms and Formulas

• General form:

• Amplitude:

• Period:

• Phase shift:

Step-by-Step Guidance

1. Identify , , and in the equation .

2. Calculate the amplitude as .

3. Find the period using .

4. Determine the phase shift using (if is present).

5. Pick five -values within one period and compute the corresponding -values for graphing.

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Q4(a). Find the exact value of .

Background

Topic: Inverse Trigonometric Functions

This question asks you to find the angle whose sine is , within the range of the arcsine function.

Key Terms and Formulas

• (arcsin) gives the angle such that and .

Step-by-Step Guidance

1. Recall the definition of and its range.

2. Think of the angle in radians whose sine is .

3. Check that this angle is within the principal range of arcsin.

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Q4(b). Find the exact value of .

Background

Topic: Inverse Trigonometric Functions

This question asks for the angle in the range of arccos whose cosine is .

Key Terms and Formulas

• (arccos) gives the angle such that and .

Step-by-Step Guidance

1. Recall the definition and range of .

2. Think of the angle in whose cosine is .

3. Check that this angle is within the principal range of arccos.

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Q4(c). Find the exact value of .

Background

Topic: Inverse Trigonometric Functions

This question asks for the angle whose tangent is $1$, within the range of arctan.

Key Terms and Formulas

• (arctan) gives the angle such that and .

Step-by-Step Guidance

1. Recall the definition and range of .

2. Think of the angle in whose tangent is $1$.

3. Check that this angle is within the principal range of arctan.

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Q5. Use your calculator to find to two decimal places.

Background

Topic: Inverse Trigonometric Functions (Calculator Use)

This question asks you to use your calculator to find the arcsine of and round to two decimal places.

Key Terms and Formulas

• returns an angle in radians (or degrees, depending on calculator mode).

• Make sure your calculator is in the correct mode (radians or degrees as required).

Step-by-Step Guidance

1. Set your calculator to radians mode (unless otherwise specified).

2. Enter into your calculator.

3. Round the result to two decimal places.

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Q6. Find .

Background

Topic: Composition of Trigonometric and Inverse Functions

This question tests your understanding of how sine and arcsine (inverse sine) interact when composed.

Key Terms and Formulas

• for in .

• Check if is in the domain of .

Step-by-Step Guidance

1. Recall the domain of is .

2. Check if is within this domain.

3. If not, consider what happens when you try to compute .

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Q7(a). Find the exact value of .

Background

Topic: Trigonometric Expressions Involving Inverse Functions

This question asks you to evaluate a sine of an arctangent, which often involves drawing a right triangle.

Key Terms and Formulas

• If , then .

• Use a right triangle to find .

Step-by-Step Guidance

1. Let , so .

2. Draw a right triangle with opposite side $1.

3. Use the Pythagorean theorem to find the hypotenuse: .

4. Express as .

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Q7(b). Find .

Background

Topic: Trigonometric Expressions Involving Inverse Functions

This question asks you to find the cotangent of an arcsine value, which can be solved using a right triangle.

Key Terms and Formulas

• If , then .

• Use a right triangle to find .

Step-by-Step Guidance

1. Let , so .

2. Draw a right triangle with opposite side $1.

3. Use the Pythagorean theorem to find the adjacent side: .

4. Express as .

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Q7(c). Find .

Background

Topic: Trigonometric Expressions Involving Inverse Functions

This question asks you to find the sine of an arccosine value. However, note that is undefined because .

Key Terms and Formulas

• The domain of is .

Step-by-Step Guidance

1. Check if is within the domain of .

2. If not, recognize that the expression is undefined.

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Q7(d). Find .

Background

Topic: Trigonometric Expressions Involving Inverse Functions

This question asks you to find the sine of an arccosine value, which can be solved using a right triangle and the Pythagorean theorem.

Key Terms and Formulas

• If , then .

• Use a right triangle to find .

Step-by-Step Guidance

1. Let , so .

2. Draw a right triangle with adjacent side and hypotenuse $3$ (since cosine is negative, the angle is in Quadrant II).

3. Use the Pythagorean theorem to find the opposite side: .

4. Express as .

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Q8. Write as an algebraic expression in terms of .

Background

Topic: Trigonometric Expressions Involving Inverse Functions

This question asks you to express a trigonometric function of an inverse trigonometric function as an algebraic expression.

Key Terms and Formulas

• If , then .

• Use a right triangle to find .

Step-by-Step Guidance

1. Let , so .

2. Draw a right triangle with opposite side and adjacent side $1$.

3. Use the Pythagorean theorem to find the hypotenuse: .

4. Express as .

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Q9. Which of the following is the shape of for between $0?

Background

Topic: Trigonometric Graphs

This question tests your ability to recognize the graph of the cosecant function over one period.

Key Terms and Formulas

• is the reciprocal of .

• It has vertical asymptotes where (at ).

• The graph consists of branches above and below the sine curve, never crossing the -axis.

Step-by-Step Guidance

1. Recall the basic shape of and where it is zero.

2. Understand that has vertical asymptotes at these zeros.

3. Visualize the branches of between the asymptotes.

4. Compare the options (A, B, C, D) and match the correct graph.

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