Trigonometry Examples

Find Trig Functions Using Identities sec(theta)=4/3 , tan(theta)<0
,
Step 1
The tangent function is negative in the second and fourth quadrants. The secant function is positive in the first and fourth quadrants. The set of solutions for are limited to the fourth quadrant since that is the only quadrant found in both sets.
Solution is in the fourth quadrant.
Step 2
Use the definition of secant 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 opposite side of the unit circle triangle. Since the adjacent side and hypotenuse 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
Negate .
Opposite
Step 5.2
Raise to the power of .
Opposite
Step 5.3
Raise to the power of .
Opposite
Step 5.4
Multiply by .
Opposite
Step 5.5
Subtract from .
Opposite
Opposite
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
Move the negative in front of the fraction.
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 8
Find the value of tangent.
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Step 8.1
Use the definition of tangent to find the value of .
Step 8.2
Substitute in the known values.
Step 8.3
Move the negative in front of the fraction.
Step 9
Find the value of cotangent.
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Step 9.1
Use the definition of cotangent to find the value of .
Step 9.2
Substitute in the known values.
Step 9.3
Simplify the value of .
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Step 9.3.1
Move the negative in front of the fraction.
Step 9.3.2
Multiply by .
Step 9.3.3
Combine and simplify the denominator.
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Step 9.3.3.1
Multiply by .
Step 9.3.3.2
Raise to the power of .
Step 9.3.3.3
Raise to the power of .
Step 9.3.3.4
Use the power rule to combine exponents.
Step 9.3.3.5
Add and .
Step 9.3.3.6
Rewrite as .
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Step 9.3.3.6.1
Use to rewrite as .
Step 9.3.3.6.2
Apply the power rule and multiply exponents, .
Step 9.3.3.6.3
Combine and .
Step 9.3.3.6.4
Cancel the common factor of .
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Step 9.3.3.6.4.1
Cancel the common factor.
Step 9.3.3.6.4.2
Rewrite the expression.
Step 9.3.3.6.5
Evaluate the exponent.
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
Simplify the value of .
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Step 10.3.1
Move the negative in front of the fraction.
Step 10.3.2
Multiply by .
Step 10.3.3
Combine and simplify the denominator.
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Step 10.3.3.1
Multiply by .
Step 10.3.3.2
Raise to the power of .
Step 10.3.3.3
Raise to the power of .
Step 10.3.3.4
Use the power rule to combine exponents.
Step 10.3.3.5
Add and .
Step 10.3.3.6
Rewrite as .
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Step 10.3.3.6.1
Use to rewrite as .
Step 10.3.3.6.2
Apply the power rule and multiply exponents, .
Step 10.3.3.6.3
Combine and .
Step 10.3.3.6.4
Cancel the common factor of .
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Step 10.3.3.6.4.1
Cancel the common factor.
Step 10.3.3.6.4.2
Rewrite the expression.
Step 10.3.3.6.5
Evaluate the exponent.
Step 11
This is the solution to each trig value.