Trigonometry Examples

Find the Sine Given the Point (5/6, square root of 11/6)
Step 1
To find the between the x-axis and the line between the points and , draw the triangle between the three points , , and .
Opposite :
Adjacent :
Step 2
Find the hypotenuse using Pythagorean theorem .
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Step 2.1
Apply the product rule to .
Step 2.2
Raise to the power of .
Step 2.3
Raise to the power of .
Step 2.4
Rewrite as .
Step 2.5
Multiply by .
Step 2.6
Combine and simplify the denominator.
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Step 2.6.1
Multiply by .
Step 2.6.2
Raise to the power of .
Step 2.6.3
Raise to the power of .
Step 2.6.4
Use the power rule to combine exponents.
Step 2.6.5
Add and .
Step 2.6.6
Rewrite as .
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Step 2.6.6.1
Use to rewrite as .
Step 2.6.6.2
Apply the power rule and multiply exponents, .
Step 2.6.6.3
Combine and .
Step 2.6.6.4
Cancel the common factor of .
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Step 2.6.6.4.1
Cancel the common factor.
Step 2.6.6.4.2
Rewrite the expression.
Step 2.6.6.5
Evaluate the exponent.
Step 2.7
Simplify the numerator.
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Step 2.7.1
Combine using the product rule for radicals.
Step 2.7.2
Multiply by .
Step 2.8
Simplify terms.
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Step 2.8.1
Apply the product rule to .
Step 2.8.2
Rewrite as .
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Step 2.8.2.1
Use to rewrite as .
Step 2.8.2.2
Apply the power rule and multiply exponents, .
Step 2.8.2.3
Combine and .
Step 2.8.2.4
Cancel the common factor of .
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Step 2.8.2.4.1
Cancel the common factor.
Step 2.8.2.4.2
Rewrite the expression.
Step 2.8.2.5
Evaluate the exponent.
Step 2.8.3
Raise to the power of .
Step 2.8.4
Cancel the common factor of and .
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Step 2.8.4.1
Factor out of .
Step 2.8.4.2
Cancel the common factors.
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Step 2.8.4.2.1
Factor out of .
Step 2.8.4.2.2
Cancel the common factor.
Step 2.8.4.2.3
Rewrite the expression.
Step 2.9
To write as a fraction with a common denominator, multiply by .
Step 2.10
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 2.10.1
Multiply by .
Step 2.10.2
Multiply by .
Step 2.11
Combine the numerators over the common denominator.
Step 2.12
Simplify the numerator.
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Step 2.12.1
Multiply by .
Step 2.12.2
Add and .
Step 2.13
Rewrite as .
Step 2.14
Simplify the denominator.
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Step 2.14.1
Rewrite as .
Step 2.14.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3
therefore .
Step 4
Simplify .
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Step 4.1
Multiply the numerator by the reciprocal of the denominator.
Step 4.2
Rewrite as .
Step 4.3
Multiply by .
Step 4.4
Combine and simplify the denominator.
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Step 4.4.1
Multiply by .
Step 4.4.2
Raise to the power of .
Step 4.4.3
Raise to the power of .
Step 4.4.4
Use the power rule to combine exponents.
Step 4.4.5
Add and .
Step 4.4.6
Rewrite as .
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Step 4.4.6.1
Use to rewrite as .
Step 4.4.6.2
Apply the power rule and multiply exponents, .
Step 4.4.6.3
Combine and .
Step 4.4.6.4
Cancel the common factor of .
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Step 4.4.6.4.1
Cancel the common factor.
Step 4.4.6.4.2
Rewrite the expression.
Step 4.4.6.5
Evaluate the exponent.
Step 4.5
Cancel the common factor of .
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Step 4.5.1
Cancel the common factor.
Step 4.5.2
Rewrite the expression.
Step 4.6
Combine using the product rule for radicals.
Step 4.7
Multiply by .
Step 4.8
Combine and .
Step 4.9
Multiply by .
Step 4.10
Combine and simplify the denominator.
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Step 4.10.1
Multiply by .
Step 4.10.2
Raise to the power of .
Step 4.10.3
Raise to the power of .
Step 4.10.4
Use the power rule to combine exponents.
Step 4.10.5
Add and .
Step 4.10.6
Rewrite as .
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Step 4.10.6.1
Use to rewrite as .
Step 4.10.6.2
Apply the power rule and multiply exponents, .
Step 4.10.6.3
Combine and .
Step 4.10.6.4
Cancel the common factor of .
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Step 4.10.6.4.1
Cancel the common factor.
Step 4.10.6.4.2
Rewrite the expression.
Step 4.10.6.5
Evaluate the exponent.
Step 4.11
Simplify the numerator.
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Step 4.11.1
Combine using the product rule for radicals.
Step 4.11.2
Multiply by .
Step 5
Approximate the result.