Trigonometry Examples

Solve the Triangle tri{5}{30}{}{60}{}{90}
Step 1
Find .
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Step 1.1
The sine of an angle is equal to the ratio of the opposite side to the hypotenuse.
Step 1.2
Substitute the name of each side into the definition of the sine function.
Step 1.3
Set up the equation to solve for the hypotenuse, in this case .
Step 1.4
Substitute the values of each variable into the formula for sine.
Step 1.5
Multiply the numerator by the reciprocal of the denominator.
Step 1.6
Multiply by .
Step 1.7
Combine and simplify the denominator.
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Step 1.7.1
Multiply by .
Step 1.7.2
Raise to the power of .
Step 1.7.3
Raise to the power of .
Step 1.7.4
Use the power rule to combine exponents.
Step 1.7.5
Add and .
Step 1.7.6
Rewrite as .
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Step 1.7.6.1
Use to rewrite as .
Step 1.7.6.2
Apply the power rule and multiply exponents, .
Step 1.7.6.3
Combine and .
Step 1.7.6.4
Cancel the common factor of .
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Step 1.7.6.4.1
Cancel the common factor.
Step 1.7.6.4.2
Rewrite the expression.
Step 1.7.6.5
Evaluate the exponent.
Step 1.8
Multiply .
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Step 1.8.1
Combine and .
Step 1.8.2
Multiply by .
Step 2
Find the last side of the triangle using the Pythagorean theorem.
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Step 2.1
Use the Pythagorean theorem to find the unknown side. In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).
Step 2.2
Solve the equation for .
Step 2.3
Substitute the actual values into the equation.
Step 2.4
Use the power rule to distribute the exponent.
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Step 2.4.1
Apply the product rule to .
Step 2.4.2
Apply the product rule to .
Step 2.5
Simplify the numerator.
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Step 2.5.1
Raise to the power of .
Step 2.5.2
Rewrite as .
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Step 2.5.2.1
Use to rewrite as .
Step 2.5.2.2
Apply the power rule and multiply exponents, .
Step 2.5.2.3
Combine and .
Step 2.5.2.4
Cancel the common factor of .
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Step 2.5.2.4.1
Cancel the common factor.
Step 2.5.2.4.2
Rewrite the expression.
Step 2.5.2.5
Evaluate the exponent.
Step 2.6
Reduce the expression by cancelling the common factors.
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Step 2.6.1
Raise to the power of .
Step 2.6.2
Multiply by .
Step 2.6.3
Cancel the common factor of and .
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Step 2.6.3.1
Factor out of .
Step 2.6.3.2
Cancel the common factors.
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Step 2.6.3.2.1
Factor out of .
Step 2.6.3.2.2
Cancel the common factor.
Step 2.6.3.2.3
Rewrite the expression.
Step 2.6.4
Simplify the expression.
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Step 2.6.4.1
Raise to the power of .
Step 2.6.4.2
Multiply by .
Step 2.7
To write as a fraction with a common denominator, multiply by .
Step 2.8
Combine and .
Step 2.9
Combine the numerators over the common denominator.
Step 2.10
Simplify the numerator.
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Step 2.10.1
Multiply by .
Step 2.10.2
Subtract from .
Step 2.11
Rewrite as .
Step 2.12
Simplify the numerator.
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Step 2.12.1
Rewrite as .
Step 2.12.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.13
Multiply by .
Step 2.14
Combine and simplify the denominator.
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Step 2.14.1
Multiply by .
Step 2.14.2
Raise to the power of .
Step 2.14.3
Raise to the power of .
Step 2.14.4
Use the power rule to combine exponents.
Step 2.14.5
Add and .
Step 2.14.6
Rewrite as .
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Step 2.14.6.1
Use to rewrite as .
Step 2.14.6.2
Apply the power rule and multiply exponents, .
Step 2.14.6.3
Combine and .
Step 2.14.6.4
Cancel the common factor of .
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Step 2.14.6.4.1
Cancel the common factor.
Step 2.14.6.4.2
Rewrite the expression.
Step 2.14.6.5
Evaluate the exponent.
Step 3
These are the results for all angles and sides for the given triangle.