Trigonometry Examples

Simplify cos(-pi/7)cos(pi/8)+sin(pi/7)sin(pi/8)
Step 1
Simplify each term.
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Step 1.1
Add full rotations of until the angle is greater than or equal to and less than .
Step 1.2
The exact value of is .
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Step 1.2.1
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Step 1.2.2
Apply the cosine half-angle identity .
Step 1.2.3
Change the to because cosine is positive in the first quadrant.
Step 1.2.4
The exact value of is .
Step 1.2.5
Simplify .
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Step 1.2.5.1
Write as a fraction with a common denominator.
Step 1.2.5.2
Combine the numerators over the common denominator.
Step 1.2.5.3
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.5.4
Multiply .
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Step 1.2.5.4.1
Multiply by .
Step 1.2.5.4.2
Multiply by .
Step 1.2.5.5
Rewrite as .
Step 1.2.5.6
Simplify the denominator.
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Step 1.2.5.6.1
Rewrite as .
Step 1.2.5.6.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.3
Combine and .
Step 1.4
The exact value of is .
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Step 1.4.1
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Step 1.4.2
Apply the sine half-angle identity.
Step 1.4.3
Change the to because sine is positive in the first quadrant.
Step 1.4.4
Simplify .
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Step 1.4.4.1
The exact value of is .
Step 1.4.4.2
Write as a fraction with a common denominator.
Step 1.4.4.3
Combine the numerators over the common denominator.
Step 1.4.4.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.4.4.5
Multiply .
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Step 1.4.4.5.1
Multiply by .
Step 1.4.4.5.2
Multiply by .
Step 1.4.4.6
Rewrite as .
Step 1.4.4.7
Simplify the denominator.
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Step 1.4.4.7.1
Rewrite as .
Step 1.4.4.7.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.5
Combine and .
Step 2
Combine the numerators over the common denominator.
Step 3
The result can be shown in multiple forms.
Exact Form:
Decimal Form: