Trigonometry Examples

Find the Cosecant Given the Point (3 1/2,-2 square root of 15)
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
Convert to an improper fraction.
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Step 2.1.1
A mixed number is an addition of its whole and fractional parts.
Step 2.1.2
Add and .
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Step 2.1.2.1
To write as a fraction with a common denominator, multiply by .
Step 2.1.2.2
Combine and .
Step 2.1.2.3
Combine the numerators over the common denominator.
Step 2.1.2.4
Simplify the numerator.
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Step 2.1.2.4.1
Multiply by .
Step 2.1.2.4.2
Add and .
Step 2.2
Simplify the expression.
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Step 2.2.1
Apply the product rule to .
Step 2.2.2
Raise to the power of .
Step 2.2.3
Raise to the power of .
Step 2.2.4
Apply the product rule to .
Step 2.2.5
Raise to the power of .
Step 2.3
Rewrite as .
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Step 2.3.1
Use to rewrite as .
Step 2.3.2
Apply the power rule and multiply exponents, .
Step 2.3.3
Combine and .
Step 2.3.4
Cancel the common factor of .
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Step 2.3.4.1
Cancel the common factor.
Step 2.3.4.2
Rewrite the expression.
Step 2.3.5
Evaluate the exponent.
Step 2.4
Multiply by .
Step 2.5
To write as a fraction with a common denominator, multiply by .
Step 2.6
Combine and .
Step 2.7
Combine the numerators over the common denominator.
Step 2.8
Simplify the numerator.
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Step 2.8.1
Multiply by .
Step 2.8.2
Add and .
Step 2.9
Rewrite as .
Step 2.10
Simplify the numerator.
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Step 2.10.1
Rewrite as .
Step 2.10.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.11
Simplify the denominator.
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Step 2.11.1
Rewrite as .
Step 2.11.2
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
Move the negative in front of the fraction.
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
Move .
Step 4.4.3
Raise to the power of .
Step 4.4.4
Raise to the power of .
Step 4.4.5
Use the power rule to combine exponents.
Step 4.4.6
Add and .
Step 4.4.7
Rewrite as .
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Step 4.4.7.1
Use to rewrite as .
Step 4.4.7.2
Apply the power rule and multiply exponents, .
Step 4.4.7.3
Combine and .
Step 4.4.7.4
Cancel the common factor of .
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Step 4.4.7.4.1
Cancel the common factor.
Step 4.4.7.4.2
Rewrite the expression.
Step 4.4.7.5
Evaluate the exponent.
Step 4.5
Multiply by .
Step 4.6
Multiply .
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Step 4.6.1
Multiply by .
Step 4.6.2
Multiply by .
Step 5
Approximate the result.