Trigonometry Examples

Solve for x csc((3pi)/4)=sin(x)
Step 1
Rewrite the equation as .
Step 2
Simplify the expression using the formula.
Step 3
Remove the from both sides of the equation.
Step 4
Since the square roots of each expression are equal, the expression inside the square root must also be equal.
Step 5
Simplify .
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Step 5.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 5.2
The exact value of is .
Step 6
Subtract from both sides of the equation.
Step 7
Multiply both sides of the equation by .
Step 8
Simplify both sides of the equation.
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Step 8.1
Simplify the left side.
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Step 8.1.1
Simplify .
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Step 8.1.1.1
Cancel the common factor of .
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Step 8.1.1.1.1
Move the leading negative in into the numerator.
Step 8.1.1.1.2
Factor out of .
Step 8.1.1.1.3
Cancel the common factor.
Step 8.1.1.1.4
Rewrite the expression.
Step 8.1.1.2
Multiply.
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Step 8.1.1.2.1
Multiply by .
Step 8.1.1.2.2
Multiply by .
Step 8.2
Simplify the right side.
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Step 8.2.1
Simplify .
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Step 8.2.1.1
Apply the distributive property.
Step 8.2.1.2
Multiply by .
Step 9
Convert the right side of the equation to its decimal equivalent.
Step 10
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 11
Simplify the right side.
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Step 11.1
Evaluate .
Step 12
Divide each term in by and simplify.
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Step 12.1
Divide each term in by .
Step 12.2
Simplify the left side.
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Step 12.2.1
Cancel the common factor of .
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Step 12.2.1.1
Cancel the common factor.
Step 12.2.1.2
Divide by .
Step 12.3
Simplify the right side.
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Step 12.3.1
Divide by .
Step 13
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 14
Solve for .
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Step 14.1
Simplify.
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Step 14.1.1
Multiply by .
Step 14.1.2
Subtract from .
Step 14.2
Divide each term in by and simplify.
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Step 14.2.1
Divide each term in by .
Step 14.2.2
Simplify the left side.
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Step 14.2.2.1
Cancel the common factor of .
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Step 14.2.2.1.1
Cancel the common factor.
Step 14.2.2.1.2
Divide by .
Step 14.2.3
Simplify the right side.
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Step 14.2.3.1
Divide by .
Step 15
Find the period of .
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Step 15.1
The period of the function can be calculated using .
Step 15.2
Replace with in the formula for period.
Step 15.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 15.4
Cancel the common factor of .
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Step 15.4.1
Cancel the common factor.
Step 15.4.2
Divide by .
Step 16
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 17
Exclude the solutions that do not make true.
No solution