# What is cos^theta in terms of sintheta?

Dec 18, 2015

If $\theta$ is in Q1 or Q4 then $\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$

If $\theta$ is in Q2 or Q3 then $\cos \theta = - \sqrt{1 - {\sin}^{2} \theta}$

#### Explanation:

We always have ${\cos}^{2} \theta + {\sin}^{2} \theta = 1$ regardless of which quadrant $\theta$ is in. Hence we always have:

$\cos \theta = \pm \sqrt{1 - {\sin}^{2} \theta}$

If $\theta$ is in Q1 or Q4 then $\cos \theta \ge 0$, hence:

$\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$

If $\theta$ is in Q2 or Q3 then $\cos \theta \le 0$, hence:

$\cos \theta = - \sqrt{1 - {\sin}^{2} \theta}$
The text passage contains a mathematical question about expressing cosine in terms of sine, and it provides a detailed answer considering different quadrants.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# What is cos^theta in terms of sintheta?

Dec 18, 2015

If $\theta$ is in Q1 or Q4 then $\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$

If $\theta$ is in Q2 or Q3 then $\cos \theta = - \sqrt{1 - {\sin}^{2} \theta}$

#### Explanation:

We always have ${\cos}^{2} \theta + {\sin}^{2} \theta = 1$ regardless of which quadrant $\theta$ is in. Hence we always have:

$\cos \theta = \pm \sqrt{1 - {\sin}^{2} \theta}$

If $\theta$ is in Q1 or Q4 then $\cos \theta \ge 0$, hence:

$\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$

If $\theta$ is in Q2 or Q3 then $\cos \theta \le 0$, hence:

$\cos \theta = - \sqrt{1 - {\sin}^{2} \theta}$
user
The text passage contains a mathematical question about expressing cosine in terms of sine, and it provides a detailed answer considering different quadrants.

TYPE: SINGLE_QUESTION_WITH_ANSWER