Trigonometry Examples

Find the Exact Value tan(112.5)
Step 1
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Step 2
Apply the tangent half-angle identity.
Step 3
Change the to because tangent is negative in the second quadrant.
Step 4
Simplify .
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Step 4.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.
Step 4.2
The exact value of is .
Step 4.3
Multiply .
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Step 4.3.1
Multiply by .
Step 4.3.2
Multiply by .
Step 4.4
Write as a fraction with a common denominator.
Step 4.5
Combine the numerators over the common denominator.
Step 4.6
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.
Step 4.7
The exact value of is .
Step 4.8
Write as a fraction with a common denominator.
Step 4.9
Combine the numerators over the common denominator.
Step 4.10
Multiply the numerator by the reciprocal of the denominator.
Step 4.11
Cancel the common factor of .
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Step 4.11.1
Cancel the common factor.
Step 4.11.2
Rewrite the expression.
Step 4.12
Multiply by .
Step 4.13
Multiply by .
Step 4.14
Expand the denominator using the FOIL method.
Step 4.15
Simplify.
Step 4.16
Apply the distributive property.
Step 4.17
Cancel the common factor of .
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Step 4.17.1
Cancel the common factor.
Step 4.17.2
Rewrite the expression.
Step 4.18
Combine and .
Step 4.19
Simplify each term.
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Step 4.19.1
Apply the distributive property.
Step 4.19.2
Move to the left of .
Step 4.19.3
Combine using the product rule for radicals.
Step 4.19.4
Simplify each term.
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Step 4.19.4.1
Multiply by .
Step 4.19.4.2
Rewrite as .
Step 4.19.4.3
Pull terms out from under the radical, assuming positive real numbers.
Step 4.19.5
Cancel the common factor of and .
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Step 4.19.5.1
Factor out of .
Step 4.19.5.2
Factor out of .
Step 4.19.5.3
Factor out of .
Step 4.19.5.4
Cancel the common factors.
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Step 4.19.5.4.1
Factor out of .
Step 4.19.5.4.2
Cancel the common factor.
Step 4.19.5.4.3
Rewrite the expression.
Step 4.19.5.4.4
Divide by .
Step 4.20
Add and .
Step 4.21
Add and .
Step 5
The result can be shown in multiple forms.
Exact Form:
Decimal Form: