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All courses Algebra II · F-TF.8 2 of 55
Prove and use sin^2(theta)+cos^2(theta)=1 to find trig values with quadrant information.

Use the Pythagorean identity to find sine when cosine and the quadrant are known

Problem
Use the \(\text{Pythagorean}~\text{identity}\) to find \(\text{sine}\) when \(\cos(\theta)=3/5\) and \(\theta\) is in \(\text{Quadrant}~I\).
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Hint

Start with \(sin^2\theta+cos^2\theta=1\) and substitute \(cos\theta=3/5\).

After solving for \(sin^2\theta\), use the quadrant to choose the correct sign of \(\sin\theta\).

Solution walkthrough

01

Apply the identity

\[\begin{aligned} \sin^2(\theta)+\cos^2(\theta)=1 \\ \sin^2(\theta)+(3/5)^2=1 \end{aligned}\]

The Pythagorean identity applies to sine and cosine of the same angle.

02

Solve the magnitude

\[\begin{aligned} \sin^2(\theta)=1-9/25=16/25 \\ |\sin(\theta)|=4/5 \end{aligned}\]

Taking a square root first gives both possible signs through the absolute value.

03

Use the quadrant

\[\text{Quadrant}~I~->~\sin(\theta)>0~->~\sin(\theta)=4/5\]

Quadrant I has positive y-coordinate and therefore positive sine, selecting the positive root. The inspected unit circle shows point (3/5,4/5).

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Another way

  1. Use a 3-4-5 right triangle with adjacent 3 and hypotenuse 5.

!

Common mistake

Do not choose -4/5 after finding the magnitude. Quadrant I requires both sine and cosine to be positive.

Answer unit-circle diagram showing the exact coordinate, quadrant signs, radius triangle, and Pythagorean identity work.
sin value: 4/5

Solution walkthrough video