Trigonometry Examples

Find the Other Trig Values in Quadrant I tan(theta) = square root of 3
Step 1
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 2
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 3
Replace the known values in the equation.
Step 4
Simplify inside the radical.
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Step 4.1
Rewrite as .
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Step 4.1.1
Use to rewrite as .
Hypotenuse
Step 4.1.2
Apply the power rule and multiply exponents, .
Hypotenuse
Step 4.1.3
Combine and .
Hypotenuse
Step 4.1.4
Cancel the common factor of .
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Step 4.1.4.1
Cancel the common factor.
Hypotenuse
Step 4.1.4.2
Rewrite the expression.
Hypotenuse
Hypotenuse
Step 4.1.5
Evaluate the exponent.
Hypotenuse
Hypotenuse
Step 4.2
One to any power is one.
Hypotenuse
Step 4.3
Add and .
Hypotenuse
Step 4.4
Rewrite as .
Hypotenuse
Step 4.5
Pull terms out from under the radical, assuming positive real numbers.
Hypotenuse
Hypotenuse
Step 5
Find the value of sine.
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Step 5.1
Use the definition of sine to find the value of .
Step 5.2
Substitute in the known values.
Step 6
Find the value of cosine.
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Step 6.1
Use the definition of cosine to find the value of .
Step 6.2
Substitute in the known values.
Step 7
Find the value of cotangent.
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Step 7.1
Use the definition of cotangent 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
Multiply by .
Step 7.3.2
Combine and simplify the denominator.
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Step 7.3.2.1
Multiply by .
Step 7.3.2.2
Raise to the power of .
Step 7.3.2.3
Raise to the power of .
Step 7.3.2.4
Use the power rule to combine exponents.
Step 7.3.2.5
Add and .
Step 7.3.2.6
Rewrite as .
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Step 7.3.2.6.1
Use to rewrite as .
Step 7.3.2.6.2
Apply the power rule and multiply exponents, .
Step 7.3.2.6.3
Combine and .
Step 7.3.2.6.4
Cancel the common factor of .
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Step 7.3.2.6.4.1
Cancel the common factor.
Step 7.3.2.6.4.2
Rewrite the expression.
Step 7.3.2.6.5
Evaluate the exponent.
Step 8
Find the value of secant.
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Step 8.1
Use the definition of secant to find the value of .
Step 8.2
Substitute in the known values.
Step 8.3
Divide by .
Step 9
Find the value of cosecant.
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Step 9.1
Use the definition of cosecant 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
Multiply by .
Step 9.3.2
Combine and simplify the denominator.
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Step 9.3.2.1
Multiply by .
Step 9.3.2.2
Raise to the power of .
Step 9.3.2.3
Raise to the power of .
Step 9.3.2.4
Use the power rule to combine exponents.
Step 9.3.2.5
Add and .
Step 9.3.2.6
Rewrite as .
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Step 9.3.2.6.1
Use to rewrite as .
Step 9.3.2.6.2
Apply the power rule and multiply exponents, .
Step 9.3.2.6.3
Combine and .
Step 9.3.2.6.4
Cancel the common factor of .
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Step 9.3.2.6.4.1
Cancel the common factor.
Step 9.3.2.6.4.2
Rewrite the expression.
Step 9.3.2.6.5
Evaluate the exponent.
Step 10
This is the solution to each trig value.