Skip to content

Physics Playground

Simulations How we teach Pricing

Language

Log in Start learning free

Loading…

Physics Playground

Built on open, peer-reviewed sources: OpenStax, The Feynman Lectures on Physics, HyperPhysics, NASA and NIST. See our sources.

Learn

Physics course
Math for Physics
Start with the math toolkit
Simulations
Glossary
Exercises
Past exams
Library
How we teach

Company

Pricing
Terms
Privacy
Contact

© 2026 Physics Playground. Made for the curious.Language

Exercises Trigonometry

Example 3 from Algebra and Trigonometry, 7.3 Unit Circle

Finding a Cosine from a Sine or a Sine from a Cosine

If $\sin (t) = \frac{3}{7}$ and $t$ is in the second quadrant, find $\cos (t) .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
If we drop a vertical line from the point on the unit circle corresponding to $t,$ we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See Figure 8.

Figure 8

Substituting the known value for sine into the Pythagorean Identity,

$\begin{matrix} {cos}^{2} (t) + \sin^{2} (t) & = & 1 \\ {cos}^{2} (t) + \frac{9}{49} & = & 1 \\ {cos}^{2} (t) & = & \frac{40}{49} \\ cos (t) & = & \pm \sqrt{\frac{40}{49}} = \pm \frac{\sqrt{40}}{7} = \pm \frac{2 \sqrt{10}}{7} \end{matrix}$

Because the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative.

$cos (t) = - \frac{2 \sqrt{10}}{7}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

If $\cos (t) = \frac{24}{25}$ and $t$ is in the fourth quadrant, find $sin (t) .$

How did it go?

PreviousCalculating Sines and Cosines along an Axis Next Using a Graphing Calculator to Find Sine and Cosine

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 7.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.