Trigonometry Examples

Simplify sin((3pi)/4)cos(pi/12)-cos((3pi)/4)sin(pi/12)
Step 1
Simplify each term.
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Step 1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.2
The exact value of is .
Step 1.3
The exact value of is .
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Step 1.3.1
Split into two angles where the values of the six trigonometric functions are known.
Step 1.3.2
Apply the difference of angles identity .
Step 1.3.3
The exact value of is .
Step 1.3.4
The exact value of is .
Step 1.3.5
The exact value of is .
Step 1.3.6
The exact value of is .
Step 1.3.7
Simplify .
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Step 1.3.7.1
Simplify each term.
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Step 1.3.7.1.1
Multiply .
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Step 1.3.7.1.1.1
Multiply by .
Step 1.3.7.1.1.2
Combine using the product rule for radicals.
Step 1.3.7.1.1.3
Multiply by .
Step 1.3.7.1.1.4
Multiply by .
Step 1.3.7.1.2
Multiply .
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Step 1.3.7.1.2.1
Multiply by .
Step 1.3.7.1.2.2
Multiply by .
Step 1.3.7.2
Combine the numerators over the common denominator.
Step 1.4
Multiply .
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Step 1.4.1
Multiply by .
Step 1.4.2
Multiply by .
Step 1.5
Apply the distributive property.
Step 1.6
Combine using the product rule for radicals.
Step 1.7
Combine using the product rule for radicals.
Step 1.8
Simplify each term.
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Step 1.8.1
Multiply by .
Step 1.8.2
Rewrite as .
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Step 1.8.2.1
Factor out of .
Step 1.8.2.2
Rewrite as .
Step 1.8.3
Pull terms out from under the radical.
Step 1.8.4
Multiply by .
Step 1.8.5
Rewrite as .
Step 1.8.6
Pull terms out from under the radical, assuming positive real numbers.
Step 1.9
Cancel the common factor of and .
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Step 1.9.1
Factor out of .
Step 1.9.2
Factor out of .
Step 1.9.3
Factor out of .
Step 1.9.4
Cancel the common factors.
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Step 1.9.4.1
Factor out of .
Step 1.9.4.2
Cancel the common factor.
Step 1.9.4.3
Rewrite the expression.
Step 1.10
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 second quadrant.
Step 1.11
The exact value of is .
Step 1.12
Multiply .
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Step 1.12.1
Multiply by .
Step 1.12.2
Multiply by .
Step 1.13
The exact value of is .
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Step 1.13.1
Split into two angles where the values of the six trigonometric functions are known.
Step 1.13.2
Apply the difference of angles identity.
Step 1.13.3
The exact value of is .
Step 1.13.4
The exact value of is .
Step 1.13.5
The exact value of is .
Step 1.13.6
The exact value of is .
Step 1.13.7
Simplify .
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Step 1.13.7.1
Simplify each term.
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Step 1.13.7.1.1
Multiply .
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Step 1.13.7.1.1.1
Multiply by .
Step 1.13.7.1.1.2
Combine using the product rule for radicals.
Step 1.13.7.1.1.3
Multiply by .
Step 1.13.7.1.1.4
Multiply by .
Step 1.13.7.1.2
Multiply .
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Step 1.13.7.1.2.1
Multiply by .
Step 1.13.7.1.2.2
Multiply by .
Step 1.13.7.2
Combine the numerators over the common denominator.
Step 1.14
Multiply .
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Step 1.14.1
Multiply by .
Step 1.14.2
Multiply by .
Step 1.15
Apply the distributive property.
Step 1.16
Combine using the product rule for radicals.
Step 1.17
Multiply .
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Step 1.17.1
Raise to the power of .
Step 1.17.2
Raise to the power of .
Step 1.17.3
Use the power rule to combine exponents.
Step 1.17.4
Add and .
Step 1.18
Simplify each term.
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Step 1.18.1
Multiply by .
Step 1.18.2
Rewrite as .
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Step 1.18.2.1
Factor out of .
Step 1.18.2.2
Rewrite as .
Step 1.18.3
Pull terms out from under the radical.
Step 1.18.4
Rewrite as .
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Step 1.18.4.1
Use to rewrite as .
Step 1.18.4.2
Apply the power rule and multiply exponents, .
Step 1.18.4.3
Combine and .
Step 1.18.4.4
Cancel the common factor of .
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Step 1.18.4.4.1
Cancel the common factor.
Step 1.18.4.4.2
Rewrite the expression.
Step 1.18.4.5
Evaluate the exponent.
Step 1.18.5
Multiply by .
Step 1.19
Cancel the common factor of and .
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Step 1.19.1
Factor out of .
Step 1.19.2
Factor out of .
Step 1.19.3
Factor out of .
Step 1.19.4
Cancel the common factors.
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Step 1.19.4.1
Factor out of .
Step 1.19.4.2
Cancel the common factor.
Step 1.19.4.3
Rewrite the expression.
Step 2
Simplify terms.
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Step 2.1
Combine the numerators over the common denominator.
Step 2.2
Add and .
Step 2.3
Simplify by subtracting numbers.
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Step 2.3.1
Subtract from .
Step 2.3.2
Add and .
Step 2.4
Cancel the common factor of and .
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Step 2.4.1
Factor out of .
Step 2.4.2
Cancel the common factors.
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Step 2.4.2.1
Factor out of .
Step 2.4.2.2
Cancel the common factor.
Step 2.4.2.3
Rewrite the expression.
Step 3
The result can be shown in multiple forms.
Exact Form:
Decimal Form: