Trigonometry Examples

Solve the Triangle tri{}{30}{10}{}{}{90}
Step 1
Find the last angle of the triangle.
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Step 1.1
The sum of all the angles in a triangle is degrees.
Step 1.2
Solve the equation for .
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Step 1.2.1
Add and .
Step 1.2.2
Move all terms not containing to the right side of the equation.
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Step 1.2.2.1
Subtract from both sides of the equation.
Step 1.2.2.2
Subtract from .
Step 2
Find .
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Step 2.1
The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.
Step 2.2
Substitute the name of each side into the definition of the cosine function.
Step 2.3
Set up the equation to solve for the adjacent side, in this case .
Step 2.4
Substitute the values of each variable into the formula for cosine.
Step 2.5
Cancel the common factor of .
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Step 2.5.1
Factor out of .
Step 2.5.2
Cancel the common factor.
Step 2.5.3
Rewrite the expression.
Step 3
Find the last side of the triangle using the Pythagorean theorem.
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Step 3.1
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 3.2
Solve the equation for .
Step 3.3
Substitute the actual values into the equation.
Step 3.4
Simplify the expression.
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Step 3.4.1
Raise to the power of .
Step 3.4.2
Apply the product rule to .
Step 3.4.3
Raise to the power of .
Step 3.5
Rewrite as .
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Step 3.5.1
Use to rewrite as .
Step 3.5.2
Apply the power rule and multiply exponents, .
Step 3.5.3
Combine and .
Step 3.5.4
Cancel the common factor of .
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Step 3.5.4.1
Cancel the common factor.
Step 3.5.4.2
Rewrite the expression.
Step 3.5.5
Evaluate the exponent.
Step 3.6
Multiply .
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Step 3.6.1
Multiply by .
Step 3.6.2
Multiply by .
Step 3.7
Subtract from .
Step 3.8
Rewrite as .
Step 3.9
Pull terms out from under the radical, assuming positive real numbers.
Step 4
These are the results for all angles and sides for the given triangle.