Trigonometry Examples

Solve for t cos(pi/2t)=-1/2
Step 1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2
Simplify the left side.
Tap for more steps...
Step 2.1
Combine and .
Step 3
Simplify the right side.
Tap for more steps...
Step 3.1
The exact value of is .
Step 4
Multiply both sides of the equation by .
Step 5
Simplify both sides of the equation.
Tap for more steps...
Step 5.1
Simplify the left side.
Tap for more steps...
Step 5.1.1
Simplify .
Tap for more steps...
Step 5.1.1.1
Cancel the common factor of .
Tap for more steps...
Step 5.1.1.1.1
Cancel the common factor.
Step 5.1.1.1.2
Rewrite the expression.
Step 5.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 5.1.1.2.1
Factor out of .
Step 5.1.1.2.2
Cancel the common factor.
Step 5.1.1.2.3
Rewrite the expression.
Step 5.2
Simplify the right side.
Tap for more steps...
Step 5.2.1
Simplify .
Tap for more steps...
Step 5.2.1.1
Cancel the common factor of .
Tap for more steps...
Step 5.2.1.1.1
Factor out of .
Step 5.2.1.1.2
Cancel the common factor.
Step 5.2.1.1.3
Rewrite the expression.
Step 5.2.1.2
Combine and .
Step 5.2.1.3
Multiply by .
Step 6
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 7
Solve for .
Tap for more steps...
Step 7.1
Multiply both sides of the equation by .
Step 7.2
Simplify both sides of the equation.
Tap for more steps...
Step 7.2.1
Simplify the left side.
Tap for more steps...
Step 7.2.1.1
Simplify .
Tap for more steps...
Step 7.2.1.1.1
Cancel the common factor of .
Tap for more steps...
Step 7.2.1.1.1.1
Cancel the common factor.
Step 7.2.1.1.1.2
Rewrite the expression.
Step 7.2.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 7.2.1.1.2.1
Factor out of .
Step 7.2.1.1.2.2
Cancel the common factor.
Step 7.2.1.1.2.3
Rewrite the expression.
Step 7.2.2
Simplify the right side.
Tap for more steps...
Step 7.2.2.1
Simplify .
Tap for more steps...
Step 7.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 7.2.2.1.2
Combine fractions.
Tap for more steps...
Step 7.2.2.1.2.1
Combine and .
Step 7.2.2.1.2.2
Combine the numerators over the common denominator.
Step 7.2.2.1.3
Simplify the numerator.
Tap for more steps...
Step 7.2.2.1.3.1
Multiply by .
Step 7.2.2.1.3.2
Subtract from .
Step 7.2.2.1.4
Simplify terms.
Tap for more steps...
Step 7.2.2.1.4.1
Cancel the common factor of .
Tap for more steps...
Step 7.2.2.1.4.1.1
Factor out of .
Step 7.2.2.1.4.1.2
Cancel the common factor.
Step 7.2.2.1.4.1.3
Rewrite the expression.
Step 7.2.2.1.4.2
Combine and .
Step 7.2.2.1.4.3
Multiply by .
Step 8
Find the period of .
Tap for more steps...
Step 8.1
The period of the function can be calculated using .
Step 8.2
Replace with in the formula for period.
Step 8.3
is approximately which is positive so remove the absolute value
Step 8.4
Multiply the numerator by the reciprocal of the denominator.
Step 8.5
Cancel the common factor of .
Tap for more steps...
Step 8.5.1
Factor out of .
Step 8.5.2
Cancel the common factor.
Step 8.5.3
Rewrite the expression.
Step 8.6
Multiply by .
Step 9
The period of the function is so values will repeat every radians in both directions.
, for any integer