Q1. Solve for all values of in the equation: . Give a general formula for all solutions.

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve basic trigonometric equations and express all possible solutions using general solution formulas.

Key Terms and Formulas:

• is the cosecant function, which is the reciprocal of :

• General solutions for trigonometric equations often involve adding integer multiples of the period (e.g., for sine/cosine, where is any integer).

Step-by-Step Guidance

1. Start by isolating in the equation: .

2. Move to the other side and solve for .

3. Recall that , so rewrite the equation in terms of .

4. Set equal to the appropriate value and determine all angles in that satisfy this equation.

5. Write the general solution for by adding integer multiples of the period.

Try solving on your own before revealing the answer!

Q2. Solve for in the equation .

Background

Topic: Solving Trigonometric Equations with Half-Angle Arguments

This question tests your understanding of solving equations involving sine with a half-angle argument and finding all solutions.

Key Terms and Formulas:

• Recall the unit circle values for .

• For , the general solution is and .

Step-by-Step Guidance

1. Let , so the equation becomes .

2. Determine all in where .

3. For each solution , solve for by multiplying both sides by 2.

4. Express the general solution for using and include all possible values by adding integer multiples of .

Try solving on your own before revealing the answer!

Q3. Solve for in the interval .

Background

Topic: Solving Basic Trigonometric Equations

This question checks your ability to isolate the trigonometric function and solve for within a specified interval.

Key Terms and Formulas:

• Inverse sine function: or

• Unit circle values for sine

Step-by-Step Guidance

1. Subtract 3 from both sides to isolate the term.

2. Divide both sides by 2 to solve for .

3. Determine all in that satisfy the resulting equation.

Try solving on your own before revealing the answer!

Q4. Solve for in the interval .

Background

Topic: Solving Trigonometric Equations

This question tests your ability to manipulate and solve equations involving cosine, including isolating the variable and using inverse trigonometric functions.

Key Terms and Formulas:

• Inverse cosine function: or

• Unit circle values for cosine

Step-by-Step Guidance

1. Subtract 2 from both sides to isolate the term.

2. Divide both sides by to solve for .

3. Find all in that satisfy the resulting equation.

Try solving on your own before revealing the answer!