Find Trig Functions Using Identities sec(theta)=2 , sin(theta)<0

Find Trig Functions Using Identities sec(theta)=2 , sin(theta)<0
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The sine function is negative in the third and fourth quadrants. The secant function is positive in the first and fourth quadrants. The set of solutions for are limited to the fourth quadrant since that is the only quadrant found in both sets.
Solution is in the fourth quadrant.
Use the definition of secant to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.
Find the opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.
Replace the known values in the equation.
Simplify inside the radical.
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Negate .
Opposite
Raise to the power of .
Opposite
One to any power is one.
Opposite
Multiply by .
Opposite
Subtract from .
Opposite
Opposite
Find the value of sine.
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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.
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Use the definition of cosine to find the value of .
Substitute in the known values.
Find the value of tangent.
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Use the definition of tangent to find the value of .
Substitute in the known values.
Divide by .
Find the value of cotangent.
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Use the definition of cotangent to find the value of .
Substitute in the known values.
Simplify the value of .
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Cancel the common factor of and .
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Rewrite as .
Move the negative in front of the fraction.
Multiply by .
Combine and simplify the denominator.
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Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Find the value of cosecant.
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Use the definition of cosecant to find the value of .
Substitute in the known values.
Simplify the value of .
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Move the negative in front of the fraction.
Multiply by .
Combine and simplify the denominator.
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Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
This is the solution to each trig value.
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Name

Name six hundred thirty million four hundred sixty-two thousand one hundred forty-seven

Interesting facts

  • 630462147 has 4 divisors, whose sum is 660484176
  • The reverse of 630462147 is 741264036
  • Previous prime number is 21

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 33
  • Digital Root 6

Name

Name one billion seven hundred forty-one million three hundred sixty thousand five hundred ninety-four

Interesting facts

  • 1741360594 has 32 divisors, whose sum is 2711326464
  • The reverse of 1741360594 is 4950631471
  • Previous prime number is 463

Basic properties

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

Name

Name two hundred six million three hundred sixty-three thousand six hundred sixty

Interesting facts

  • 206363660 has 32 divisors, whose sum is 570773952
  • The reverse of 206363660 is 066363602
  • Previous prime number is 41

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 32
  • Digital Root 5