Find Trig Functions Using Identities csc(theta)=4 , cot(theta)<0

Find Trig Functions Using Identities csc(theta)=4 , cot(theta)<0
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The cotangent function is negative in the second and fourth quadrants. The cosecant function is positive in the first and second 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 cosecant to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.
Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.
Replace the known values in the equation.
Negate .
Raise to the power of .
One to any power is one.
Multiply by .
Subtract from .
Find the value of sine.
Use the definition of sine to find the value of .
Substitute in the known values.
Find the value of cosine.
Use the definition of cosine to find the value of .
Substitute in the known values.
Move the negative in front of the fraction.
Find the value of tangent.
Use the definition of tangent to find the value of .
Substitute in the known values.
Simplify the value of .
Cancel the common factor of and .
Rewrite as .
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.
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.
This is the solution to each trig value.
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Name

Name two billion twenty-three million seven hundred ninety thousand thirty-eight

Interesting facts

• 2023790038 has 4 divisors, whose sum is 3035685060
• The reverse of 2023790038 is 8300973202
• Previous prime number is 2

Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 34
• Digital Root 7

Name

Name forty-three million nine hundred twenty-six thousand eight hundred twenty-six

Interesting facts

• 43926826 has 16 divisors, whose sum is 70435008
• The reverse of 43926826 is 62862934
• Previous prime number is 23

Basic properties

• Is Prime? no
• Number parity even
• Number length 8
• Sum of Digits 40
• Digital Root 4

Name

Name two billion thirty-two million four hundred seventy-four thousand seven hundred fifty-eight

Interesting facts

• 2032474758 has 16 divisors, whose sum is 4079627760
• The reverse of 2032474758 is 8574742302
• Previous prime number is 277

Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 42
• Digital Root 6