Trigonometry Examples

Solve for x sin(x)^2+8sin(x)-4=0
Step 1
Substitute for .
Step 2
Use the quadratic formula to find the solutions.
Step 3
Substitute the values , , and into the quadratic formula and solve for .
Step 4
Simplify.
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Step 4.1
Simplify the numerator.
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Step 4.1.1
Raise to the power of .
Step 4.1.2
Multiply .
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Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply by .
Step 4.1.3
Add and .
Step 4.1.4
Rewrite as .
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Step 4.1.4.1
Factor out of .
Step 4.1.4.2
Rewrite as .
Step 4.1.5
Pull terms out from under the radical.
Step 4.2
Multiply by .
Step 4.3
Simplify .
Step 5
The final answer is the combination of both solutions.
Step 6
Substitute for .
Step 7
Set up each of the solutions to solve for .
Step 8
Solve for in .
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Step 8.1
Convert the right side of the equation to its decimal equivalent.
Step 8.2
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 8.3
Simplify the right side.
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Step 8.3.1
Evaluate .
Step 8.4
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 8.5
Solve for .
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Step 8.5.1
Remove parentheses.
Step 8.5.2
Remove parentheses.
Step 8.5.3
Subtract from .
Step 8.6
Find the period of .
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Step 8.6.1
The period of the function can be calculated using .
Step 8.6.2
Replace with in the formula for period.
Step 8.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 8.6.4
Divide by .
Step 8.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 9
Solve for in .
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Step 9.1
Convert the right side of the equation to its decimal equivalent.
Step 9.2
The range of sine is . Since does not fall in this range, there is no solution.
No solution
No solution
Step 10
List all of the solutions.
, for any integer