Find Trig Functions Using Identities tan(theta)=24/7 , sin(theta)<0

Find Trig Functions Using Identities tan(theta)=24/7 , 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
Rewrite as .
Hypotenuse
Pull terms out from under the radical, assuming positive real numbers.
Hypotenuse
Hypotenuse
Find the value of sine.
Use the definition of sine to find the value of .
Substitute in the known values.
Move the negative in front of the fraction.
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 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.
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.
Move the negative in front of the fraction.
This is the solution to each trig value.
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Name

Name one billion nine hundred fifteen million seven hundred fifty-two thousand two hundred twenty-four

Interesting facts

• 1915752224 has 128 divisors, whose sum is 14902576668
• The reverse of 1915752224 is 4222575191
• Previous prime number is 41

Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 38
• Digital Root 2

Name

Name one billion three hundred fifty-nine million fifty-seven thousand two hundred ninety-three

Interesting facts

• 1359057293 has 8 divisors, whose sum is 1497003456
• The reverse of 1359057293 is 3927509531
• Previous prime number is 103

Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 44
• Digital Root 8

Name

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

Interesting facts

• 1020877254 has 16 divisors, whose sum is 1703473200
• The reverse of 1020877254 is 4527780201
• Previous prime number is 857

Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 36
• Digital Root 9