Find Trig Functions Using Identities tan(theta)=-1/3 , sin(theta)>0

Find Trig Functions Using Identities tan(theta)=-1/3 , sin(theta)>0
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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.
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.
One to any power is one.
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 .
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.
Divide by .
Find the value of secant.
Use the definition of secant to find the value of .
Substitute in the known values.
Move the negative in front of the fraction.
Find the value of cosecant.
Use the definition of cosecant to find the value of .
Substitute in the known values.
Divide by .
This is the solution to each trig value.
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Name

Name one billion six hundred fifty million eight hundred twenty-six thousand six hundred eighty-five

Interesting facts

• 1650826685 has 16 divisors, whose sum is 2086930560
• The reverse of 1650826685 is 5866280561
• Previous prime number is 12583

Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 47
• Digital Root 2

Name

Name six hundred seventy-two million two hundred thirty-six thousand nine hundred seventy-one

Interesting facts

• 672236971 has 8 divisors, whose sum is 768526304
• The reverse of 672236971 is 179632276
• Previous prime number is 3361

Basic properties

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

Name

Name one billion six hundred thirty-four million nine hundred seventy-one thousand two hundred four

Interesting facts

• 1634971204 has 32 divisors, whose sum is 3874068000
• The reverse of 1634971204 is 4021794361
• Previous prime number is 3299

Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 37
• Digital Root 1