Trigonometry Examples

Simplify sin(165)+sin(105)
Step 1
Simplify each term.
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Step 1.1
The exact value of is .
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Step 1.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.1.2
Split into two angles where the values of the six trigonometric functions are known.
Step 1.1.3
Separate negation.
Step 1.1.4
Apply the difference of angles identity.
Step 1.1.5
The exact value of is .
Step 1.1.6
The exact value of is .
Step 1.1.7
The exact value of is .
Step 1.1.8
The exact value of is .
Step 1.1.9
Simplify .
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Step 1.1.9.1
Simplify each term.
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Step 1.1.9.1.1
Multiply .
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Step 1.1.9.1.1.1
Multiply by .
Step 1.1.9.1.1.2
Combine using the product rule for radicals.
Step 1.1.9.1.1.3
Multiply by .
Step 1.1.9.1.1.4
Multiply by .
Step 1.1.9.1.2
Multiply .
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Step 1.1.9.1.2.1
Multiply by .
Step 1.1.9.1.2.2
Multiply by .
Step 1.1.9.2
Combine the numerators over the common denominator.
Step 1.2
The exact value of is .
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Step 1.2.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.2.2
Split into two angles where the values of the six trigonometric functions are known.
Step 1.2.3
Apply the sum of angles identity.
Step 1.2.4
The exact value of is .
Step 1.2.5
The exact value of is .
Step 1.2.6
The exact value of is .
Step 1.2.7
The exact value of is .
Step 1.2.8
Simplify .
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Step 1.2.8.1
Simplify each term.
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Step 1.2.8.1.1
Multiply .
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Step 1.2.8.1.1.1
Multiply by .
Step 1.2.8.1.1.2
Multiply by .
Step 1.2.8.1.2
Multiply .
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Step 1.2.8.1.2.1
Multiply by .
Step 1.2.8.1.2.2
Combine using the product rule for radicals.
Step 1.2.8.1.2.3
Multiply by .
Step 1.2.8.1.2.4
Multiply by .
Step 1.2.8.2
Combine the numerators over the common denominator.
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
Add and .
Step 2.4
Add and .
Step 2.5
Cancel the common factor of and .
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Step 2.5.1
Factor out of .
Step 2.5.2
Cancel the common factors.
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Step 2.5.2.1
Factor out of .
Step 2.5.2.2
Cancel the common factor.
Step 2.5.2.3
Rewrite the expression.
Step 3
The result can be shown in multiple forms.
Exact Form:
Decimal Form: