Q1. Establish each identity.

Background

Topic: Trigonometric Identities

This question tests your understanding of fundamental and derived trigonometric identities. You are expected to manipulate one side of the equation to show it is equivalent to the other, using algebraic and trigonometric properties.

Key Terms and Formulas:

• Trigonometric identities (e.g., Pythagorean, reciprocal, quotient, co-function, sum/difference, double-angle, half-angle)

• Algebraic manipulation

Step-by-Step Guidance

1. Identify which side of the identity is more complex and start manipulating that side.

2. Recall relevant identities (e.g., , , etc.).

3. Substitute or rewrite terms using these identities to simplify the expression.

4. Combine like terms and simplify algebraically, aiming to match the other side of the equation.

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Q2. Solve the equation. Give a general formula for all the solutions.

Background

Topic: Solving Trigonometric Equations

This question assesses your ability to solve trigonometric equations and express all possible solutions using a general formula (often involving or for integer ).

Key Terms and Formulas:

• Inverse trigonometric functions

• General solution for , ,

• Periodicity of trigonometric functions

Step-by-Step Guidance

1. Isolate the trigonometric function (e.g., , , or ) on one side of the equation.

2. Apply the appropriate inverse function to both sides to solve for .

3. Recall the general solution for the function (e.g., or for integer ).

4. Write the general solution, making sure to include all possible solutions within the period of the function.

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Q3. Find critical number(s) of on the interval .

Background

Topic: Critical Numbers and Extrema

This question tests your ability to find critical numbers of a function, which are points where the derivative is zero or undefined, within a specified interval.

Key Terms and Formulas:

• Critical number: A value in the domain of where or does not exist.

• Derivative

Step-by-Step Guidance

1. Compute the derivative of the given function.

2. Set and solve for within the interval .

3. Check for values where does not exist, if applicable, and include those in your list if they are in the interval.

4. List all critical numbers found in the interval.

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Q4. Solve the equation on the interval .

Background

Topic: Solving Trigonometric Equations on a Restricted Interval

This question asks you to find all solutions to a trigonometric equation within a specific interval, typically .

Key Terms and Formulas:

• Trigonometric equations

• Unit circle values

• Interval notation

Step-by-Step Guidance

1. Isolate the trigonometric function in the equation.

2. Use inverse trigonometric functions to find possible solutions for .

3. Check all solutions within the interval , considering the periodicity and symmetry of the function.

4. List all solutions that satisfy the equation in the given interval.

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Q5. Find the exact value of given that ...

Background

Topic: Trigonometric Angle Difference Formulas

This question tests your ability to use the sine difference formula and given values to compute an exact trigonometric value.

Key Terms and Formulas:

• Sine difference formula:

• Exact values (no decimals)

Step-by-Step Guidance

1. Write out the sine difference formula.

2. Substitute the given values for , , , and as provided in the problem.

3. Simplify the expression, keeping terms in exact form (e.g., in terms of , , etc.).

4. Combine like terms as much as possible, but do not convert to decimals.

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Q6. Find the exact value of each expression.

Background

Topic: Evaluating Trigonometric Expressions

This question asks you to find the exact value of given trigonometric expressions, likely at special angles or using identities.

Key Terms and Formulas:

• Unit circle values

• Special angles (e.g., , etc.)

• Relevant identities (if needed)

Step-by-Step Guidance

1. Identify the angle or expression to be evaluated.

2. Recall the exact value for the trigonometric function at that angle, or use identities to simplify if necessary.

3. Keep your answer in exact form (e.g., , , etc.).

4. Double-check your answer using the unit circle or known values.

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Q7. Compute the derivative of each function.

Background

Topic: Differentiation

This question tests your ability to compute derivatives using rules such as the power rule, product rule, quotient rule, and chain rule.

Key Terms and Formulas:

• Power rule:

• Product rule:

• Quotient rule:

• Chain rule:

Step-by-Step Guidance

1. Identify which differentiation rule(s) apply to the given function.

2. Apply the rule(s) step by step, writing out each derivative component.

3. Simplify the resulting expression as much as possible, but do not combine into a final simplified answer yet.

4. Check your work for algebraic accuracy and correct application of the rules.

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