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 hundred thirty-two million four hundred fifty-eight thousand eight hundred ninety-nine

Interesting facts

• 132458899 has 8 divisors, whose sum is 139628880
• The reverse of 132458899 is 998854231
• Previous prime number is 761

Basic properties

• Is Prime? no
• Number parity odd
• Number length 9
• Sum of Digits 49
• Digital Root 4

Name

Name two hundred thirty-four million three hundred twenty-eight thousand seven hundred eighty-three

Interesting facts

• 234328783 has 16 divisors, whose sum is 261216256
• The reverse of 234328783 is 387823432
• Previous prime number is 1063

Basic properties

• Is Prime? no
• Number parity odd
• Number length 9
• Sum of Digits 40
• Digital Root 4

Name

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

Interesting facts

• 1689222763 has 4 divisors, whose sum is 1705623088
• The reverse of 1689222763 is 3672229861
• Previous prime number is 103

Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 46
• Digital Root 1