Trigonometry Examples

Solve for t 16=-10cos((2pi)/150t)+10
Step 1
Rewrite the equation as .
Step 2
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 2.1
Subtract from both sides of the equation.
Step 2.2
Subtract from .
Step 3
Divide each term in by and simplify.
Tap for more steps...
Step 3.1
Divide each term in by .
Step 3.2
Simplify the left side.
Tap for more steps...
Step 3.2.1
Cancel the common factor of .
Tap for more steps...
Step 3.2.1.1
Cancel the common factor.
Step 3.2.1.2
Divide by .
Step 3.3
Simplify the right side.
Tap for more steps...
Step 3.3.1
Cancel the common factor of and .
Tap for more steps...
Step 3.3.1.1
Factor out of .
Step 3.3.1.2
Cancel the common factors.
Tap for more steps...
Step 3.3.1.2.1
Factor out of .
Step 3.3.1.2.2
Cancel the common factor.
Step 3.3.1.2.3
Rewrite the expression.
Step 3.3.2
Move the negative in front of the fraction.
Step 4
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 5
Simplify the left side.
Tap for more steps...
Step 5.1
Simplify .
Tap for more steps...
Step 5.1.1
Cancel the common factor of and .
Tap for more steps...
Step 5.1.1.1
Factor out of .
Step 5.1.1.2
Cancel the common factors.
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 5.1.2
Combine and .
Step 6
Simplify the right side.
Tap for more steps...
Step 6.1
Evaluate .
Step 7
Multiply both sides of the equation by .
Step 8
Simplify both sides of the equation.
Tap for more steps...
Step 8.1
Simplify the left side.
Tap for more steps...
Step 8.1.1
Simplify .
Tap for more steps...
Step 8.1.1.1
Cancel the common factor of .
Tap for more steps...
Step 8.1.1.1.1
Cancel the common factor.
Step 8.1.1.1.2
Rewrite the expression.
Step 8.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 8.1.1.2.1
Factor out of .
Step 8.1.1.2.2
Cancel the common factor.
Step 8.1.1.2.3
Rewrite the expression.
Step 8.2
Simplify the right side.
Tap for more steps...
Step 8.2.1
Simplify .
Tap for more steps...
Step 8.2.1.1
Multiply .
Tap for more steps...
Step 8.2.1.1.1
Combine and .
Step 8.2.1.1.2
Multiply by .
Step 8.2.1.2
Replace with an approximation.
Step 8.2.1.3
Divide by .
Step 9
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 10
Solve for .
Tap for more steps...
Step 10.1
Multiply both sides of the equation by .
Step 10.2
Simplify both sides of the equation.
Tap for more steps...
Step 10.2.1
Simplify the left side.
Tap for more steps...
Step 10.2.1.1
Simplify .
Tap for more steps...
Step 10.2.1.1.1
Cancel the common factor of .
Tap for more steps...
Step 10.2.1.1.1.1
Cancel the common factor.
Step 10.2.1.1.1.2
Rewrite the expression.
Step 10.2.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 10.2.1.1.2.1
Factor out of .
Step 10.2.1.1.2.2
Cancel the common factor.
Step 10.2.1.1.2.3
Rewrite the expression.
Step 10.2.2
Simplify the right side.
Tap for more steps...
Step 10.2.2.1
Simplify .
Tap for more steps...
Step 10.2.2.1.1
Multiply by .
Step 10.2.2.1.2
Subtract from .
Step 10.2.2.1.3
Multiply .
Tap for more steps...
Step 10.2.2.1.3.1
Combine and .
Step 10.2.2.1.3.2
Multiply by .
Step 10.2.2.1.4
Replace with an approximation.
Step 10.2.2.1.5
Divide by .
Step 11
Find the period of .
Tap for more steps...
Step 11.1
The period of the function can be calculated using .
Step 11.2
Replace with in the formula for period.
Step 11.3
is approximately which is positive so remove the absolute value
Step 11.4
Multiply the numerator by the reciprocal of the denominator.
Step 11.5
Cancel the common factor of .
Tap for more steps...
Step 11.5.1
Factor out of .
Step 11.5.2
Cancel the common factor.
Step 11.5.3
Rewrite the expression.
Step 11.6
Multiply by .
Step 12
The period of the function is so values will repeat every radians in both directions.
, for any integer