Trigonometry Examples

Find Trig Functions Using Identities tan(theta)=-1/3 , sin(theta)>0
,
Step 1
The sine function is positive in the first and second quadrants. The tangent function is negative in the second and fourth quadrants. The set of solutions for are limited to the second quadrant since that is the only quadrant found in both sets.
Solution is in the second quadrant.
Step 2
Use the definition of tangent to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.
Step 3
Find the hypotenuse of the unit circle triangle. Since the opposite and adjacent sides are known, use the Pythagorean theorem to find the remaining side.
Step 4
Replace the known values in the equation.
Step 5
Simplify inside the radical.
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Step 5.1
One to any power is one.
Hypotenuse
Step 5.2
Raise to the power of .
Hypotenuse
Step 5.3
Add and .
Hypotenuse
Hypotenuse
Step 6
Find the value of sine.
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Step 6.1
Use the definition of sine to find the value of .
Step 6.2
Substitute in the known values.
Step 6.3
Simplify the value of .
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Step 6.3.1
Multiply by .
Step 6.3.2
Combine and simplify the denominator.
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Step 6.3.2.1
Multiply by .
Step 6.3.2.2
Raise to the power of .
Step 6.3.2.3
Raise to the power of .
Step 6.3.2.4
Use the power rule to combine exponents.
Step 6.3.2.5
Add and .
Step 6.3.2.6
Rewrite as .
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Step 6.3.2.6.1
Use to rewrite as .
Step 6.3.2.6.2
Apply the power rule and multiply exponents, .
Step 6.3.2.6.3
Combine and .
Step 6.3.2.6.4
Cancel the common factor of .
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Step 6.3.2.6.4.1
Cancel the common factor.
Step 6.3.2.6.4.2
Rewrite the expression.
Step 6.3.2.6.5
Evaluate the exponent.
Step 7
Find the value of cosine.
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Step 7.1
Use the definition of cosine to find the value of .
Step 7.2
Substitute in the known values.
Step 7.3
Simplify the value of .
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Step 7.3.1
Move the negative in front of the fraction.
Step 7.3.2
Multiply by .
Step 7.3.3
Combine and simplify the denominator.
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Step 7.3.3.1
Multiply by .
Step 7.3.3.2
Raise to the power of .
Step 7.3.3.3
Raise to the power of .
Step 7.3.3.4
Use the power rule to combine exponents.
Step 7.3.3.5
Add and .
Step 7.3.3.6
Rewrite as .
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Step 7.3.3.6.1
Use to rewrite as .
Step 7.3.3.6.2
Apply the power rule and multiply exponents, .
Step 7.3.3.6.3
Combine and .
Step 7.3.3.6.4
Cancel the common factor of .
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Step 7.3.3.6.4.1
Cancel the common factor.
Step 7.3.3.6.4.2
Rewrite the expression.
Step 7.3.3.6.5
Evaluate the exponent.
Step 8
Find the value of cotangent.
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Step 8.1
Use the definition of cotangent to find the value of .
Step 8.2
Substitute in the known values.
Step 8.3
Divide by .
Step 9
Find the value of secant.
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Step 9.1
Use the definition of secant to find the value of .
Step 9.2
Substitute in the known values.
Step 9.3
Move the negative in front of the fraction.
Step 10
Find the value of cosecant.
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Step 10.1
Use the definition of cosecant to find the value of .
Step 10.2
Substitute in the known values.
Step 10.3
Divide by .
Step 11
This is the solution to each trig value.