# Find Trig Functions Using Identities tan(theta)=2 , sin(theta)<0

Find Trig Functions Using Identities tan(theta)=2 , sin(theta)<0
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The sine function is negative in the third and fourth quadrants. The tangent function is positive in the first and third quadrants. The set of solutions for are limited to the third quadrant since that is the only quadrant found in both sets.
Solution is in the third quadrant.
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.
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.
Replace the known values in the equation.
Raise to the power of .
Hypotenuse
Raise to the power of .
Hypotenuse
Hypotenuse
Hypotenuse
Find the value of sine.
Use the definition of sine to find the value of .
Substitute in the known values.
Simplify the value of .
Move the negative in front of the fraction.
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Find the value of cosine.
Use the definition of cosine to find the value of .
Substitute in the known values.
Simplify the value of .
Move the negative in front of the fraction.
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Find the value of cotangent.
Use the definition of cotangent to find the value of .
Substitute in the known values.
Dividing two negative values results in a positive value.
Find the value of secant.
Use the definition of secant to find the value of .
Substitute in the known values.
Simplify the value of .
Move the negative one from the denominator of .
Rewrite as .
Find the value of cosecant.
Use the definition of cosecant to find the value of .
Substitute in the known values.
Move the negative in front of the fraction.
This is the solution to each trig value.
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### Name

Name eighty-two million twenty-eight thousand seven hundred ten

### Interesting facts

• 82028710 has 8 divisors, whose sum is 147651696
• The reverse of 82028710 is 01782028
• Previous prime number is 5

### Basic properties

• Is Prime? no
• Number parity even
• Number length 8
• Sum of Digits 28
• Digital Root 1

### Name

Name one billion nine hundred seven million three hundred twenty-two thousand seven hundred eighty-four

### Interesting facts

• 1907322784 has 128 divisors, whose sum is 14627160468
• The reverse of 1907322784 is 4872237091
• Previous prime number is 101

### Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 43
• Digital Root 7

### Name

Name five hundred thirty-seven million eight hundred forty-two thousand five hundred eighteen

### Interesting facts

• 537842518 has 8 divisors, whose sum is 806888520
• The reverse of 537842518 is 815248735
• Previous prime number is 8011

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 43
• Digital Root 7