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 <math><mstyle displaystyle="true"><mi>θ</mi></mstyle></math> 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.

Negate <math><mstyle displaystyle="true"><msqrt><msup><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>1</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><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><msup><mrow><mo>(</mo><mn>1</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><mo>-</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

One to any power is one.

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><mo>-</mo><mn>1</mn><mo>⋅</mo><mn>1</mn></msqrt></mstyle></math>

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

Opposite <math><mstyle displaystyle="true"><mo>=</mo><mo>-</mo><msqrt><mn>4</mn><mo>-</mo><mn>1</mn></msqrt></mstyle></math>

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>4</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.

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.

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.

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

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> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> .

Move the negative in front of the fraction.

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</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>1</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> .

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.

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> .

Move the negative in front of the fraction.

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

This is the solution to each trig value.

Do you know how to Find Trig Functions Using Identities sec(theta)=2 , sin(theta)<0? 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 | six hundred thirty million four hundred sixty-two thousand one hundred forty-seven |
---|

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

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

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

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

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

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

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

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