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# Trigonometry Examples

To find the x-intercept, substitute in for and solve for .

Rewrite the equation as .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Take the inverse cosine of both sides of the equation to extract from inside the cosine.

The exact value of is .

Move all terms not containing to the right side of the equation.

Add to both sides of the equation.

Simplify the right side of the equation.

Combine the numerators over the common denominator.

Add and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.

Simplify the expression to find the second solution.

Simplify .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

Move all terms not containing to the right side of the equation.

Add to both sides of the equation.

Simplify the right side of the equation.

Combine the numerators over the common denominator.

Add and .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Find the period.

The period of the function can be calculated using .

Replace with in the formula for period.

Solve the equation.

The absolute value is the distance between a number and zero. The distance between and is .

Divide by .

The period of the function is so values will repeat every radians in both directions.

, for any integer

Consolidate the answers.

, for any integer

, for any integer

To find the y-intercept, substitute in for and solve for .

Subtract from .

Add full rotations of until the angle is between and .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

The exact value of is .

Multiply by .

These are the and intercepts of the equation .

x-intercept: , for any integer

y-intercept: