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.

Negate <math><mstyle displaystyle="true"><msqrt><msup><mrow><mo>(</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><msup><mrow><mo>(</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Apply the product rule to <math><mstyle displaystyle="true"><mn>2</mn><msqrt><mn>3</mn></msqrt></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Raise <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><msup><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Rewrite <math><mstyle displaystyle="true"><msup><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Use <math><mstyle displaystyle="true"><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>n</mi></mrow></mroot><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup></mstyle></math> to rewrite <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><msup><mrow><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><mo>⋅</mo><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><mo>⋅</mo><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><mo>⋅</mo><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><menclose notation="updiagonalstrike"><mn>2</mn></menclose></mrow><mrow><menclose notation="updiagonalstrike"><mn>2</mn></menclose></mrow></mfrac></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Rewrite the expression.

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><mo>⋅</mo><mn>3</mn><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><mo>⋅</mo><mn>3</mn><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Evaluate the exponent.

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><mo>⋅</mo><mn>3</mn><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Multiply <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>12</mn><mo>-</mo><msup><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Raise <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>12</mn><mo>-</mo><mn>1</mn><mo>⋅</mo><mn>9</mn></msqrt></mstyle></math>

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>12</mn><mo>-</mo><mn>9</mn></msqrt></mstyle></math>

Subtract <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>12</mn></mstyle></math> .

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>3</mn></msqrt></mstyle></math>

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>3</mn></msqrt></mstyle></math>

Use the definition of sine to find the value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Simplify the value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Use the definition of cosine to find the value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Simplify the value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mstyle></math> .

Combine and simplify the denominator.

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mstyle></math> .

Move <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Use <math><mstyle displaystyle="true"><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>n</mi></mrow></mroot><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup></mstyle></math> to rewrite <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Evaluate the exponent.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Use the definition of tangent to find the value of <math><mstyle displaystyle="true"><mi>tan</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Move the negative in front of the fraction.

Use the definition of cotangent to find the value of <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Simplify the value of <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Move the negative in front of the fraction.

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mstyle></math> .

Combine and simplify the denominator.

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Use <math><mstyle displaystyle="true"><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>n</mi></mrow></mroot><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup></mstyle></math> to rewrite <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Evaluate the exponent.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Use the definition of cosecant to find the value of <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Simplify the value of <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Move the negative one from the denominator of <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

This is the solution to each trig value.

Do you know how to Find the Other Trig Values in Quadrant IV sec(theta)=(2 square root of 3)/3? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name | one billion eight hundred three million three hundred seventy-three thousand five hundred forty-two |
---|

- 1803373542 has 16 divisors, whose sum is
**4808996160** - The reverse of 1803373542 is
**2453733081** - Previous prime number is
**3**

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

Name | one billion five hundred seventy-seven million four hundred three thousand five hundred ten |
---|

- 1577403510 has 16 divisors, whose sum is
**2538966528** - The reverse of 1577403510 is
**0153047751** - Previous prime number is
**167**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 33
- Digital Root 6

Name | seven hundred thirty-seven million six hundred thousand three hundred twenty-four |
---|

- 737600324 has 16 divisors, whose sum is
**1659901248** - The reverse of 737600324 is
**423006737** - Previous prime number is
**6983**

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