Chapter 12.3, Problem 51HP

To determine

To find: Describe the steps for evaluating a trigonometric function for an angle $\theta$ that is greater than ${90}^0$ .

Explanation of Solution

Given information:

Angle $\theta$ that is greater than ${90}^0$ .

Concept:

If $\theta$ is the non-quadrantal angle in standard position, its reference angle ${\theta }^{\text{'}}$ is the acute angle formed by the terminal side of $\theta$ and the x-axis.

The ruled for finding the measures of reference angles for $0^0<\theta <{360}^0$ or $0^0<\theta <2\pi$ are shown

For Quadrant − I

  ${\theta }^{\text{'}}=\theta$

For Quadrant − II

  ${\theta }^{\text{'}}={180}^0-\theta$ or ${\theta }^{\text{'}}=\pi -\theta$

For Quadrant − III

  ${\theta }^{\text{'}}=\theta -{180}^0$ or ${\theta }^{\text{'}}=\theta -\pi$

For Quadrant − IV

  ${\theta }^{\text{'}}={360}^0-\theta$ or ${\theta }^{\text{'}}=2\pi -\theta$

Given

Angle $\theta$ that is greater than ${90}^0$

It means $\theta$ should be in Quadrant-II or in Quadrant-III or in Quadrant-IV

If we it is in Quadrant-II

Then we need to calculate reference angle using

For Quadrant − II

  ${\theta }^{\text{'}}={180}^0-\theta$ or ${\theta }^{\text{'}}=\pi -\theta$

Then we need to calculate value of Trigonometric function using that reference angle.

Using in Quadrant-II only Sine and Cosecant functions are Positive Condition.

If we it is in Quadrant-III

Then we need to calculate reference angle using

For Quadrant − III

  ${\theta }^{\text{'}}=\theta -{180}^0$ or ${\theta }^{\text{'}}=\theta -\pi$

Then we need to calculate value of Trigonometric function using that reference angle.

Using in Quadrant-III only Tan and Cot functions are Positive Condition.

If we it is in Quadrant-IV

Then we need to calculate reference angle using

For Quadrant − IV

  ${\theta }^{\text{'}}={360}^0-\theta$ or ${\theta }^{\text{'}}=2\pi -\theta$

Then we need to calculate value of Trigonometric function using that reference angle.

Using in Quadrant-IV only Cosine and Secant functions are Positive Condition.