# 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 nine hundred six million one hundred forty-seven thousand five hundred twenty-eight

### Interesting facts

• 906147528 has 128 divisors, whose sum is 4790560320
• The reverse of 906147528 is 825741609
• Previous prime number is 13

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 42
• Digital Root 6

### Name

Name two billion sixty-three million two hundred ninety-three thousand two hundred eighty-five

### Interesting facts

• 2063293285 has 4 divisors, whose sum is 2475951948
• The reverse of 2063293285 is 5823923602
• Previous prime number is 5

### Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 40
• Digital Root 4

### Name

Name two hundred forty-five million one hundred twenty-eight thousand five hundred forty

### Interesting facts

• 245128540 has 16 divisors, whose sum is 661847112
• The reverse of 245128540 is 045821542
• Previous prime number is 5

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 31
• Digital Root 4