Find Trig Functions Using Identities sin(theta)=2/3 , tan(theta)<0

Find Trig Functions Using Identities sin(theta)=2/3 , tan(theta)<0
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The tangent function is negative in the second and fourth quadrants. The sine 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 sine 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.
Simplify inside the radical.
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Negate .
Adjacent
Raise to the power of .
Adjacent
Raise to the power of .
Adjacent
Multiply by .
Adjacent
Subtract from .
Adjacent
Adjacent
Find the value of cosine.
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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.
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Use the definition of tangent 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.
Find the value of cotangent.
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Use the definition of cotangent to find the value of .
Substitute in the known values.
Move the negative in front of the fraction.
Find the value of secant.
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Use the definition of secant 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.
Find the value of cosecant.
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Use the definition of cosecant to find the value of .
Substitute in the known values.
This is the solution to each trig value.
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Name

Name one billion nine hundred ninety-five million seventy-two thousand eight hundred eighty

Interesting facts

  • 1995072880 has 128 divisors, whose sum is 12127072464
  • The reverse of 1995072880 is 0882705991
  • Previous prime number is 2011

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 49
  • Digital Root 4

Name

Name five hundred sixteen million four hundred sixty-five thousand nine hundred twenty-six

Interesting facts

  • 516465926 has 4 divisors, whose sum is 774698892
  • The reverse of 516465926 is 629564615
  • Previous prime number is 2

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 44
  • Digital Root 8

Name

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

Interesting facts

  • 1089230793 has 4 divisors, whose sum is 1141098948
  • The reverse of 1089230793 is 3970329801
  • Previous prime number is 21

Basic properties

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