Guided Learning Extension
Given a coordinate, find trig ratios.
Example 1: The coordinate \(\left(2\sqrt{3},-5\right)\) lies on the terminal side of \(\theta\). Find the exact values of \(\sin\theta\) and \(\cot\theta\).
Begin by sketching a picture.
Using Pythagorean theorem, we can find the hypotenuse of the right triangle with side lengths of \(5\) and \(2\sqrt{3}\).
\(\left(2\sqrt{3}\right)^2+5^2=c^2\)
\(12+25=c^2\)
\(c=\sqrt{37}\)
Once you know all three sides of the triangle, you can use them to write any trig ratio.
\(\sin\theta=\large-\frac{5}{\sqrt{37}}\)
\(\sin\theta=\large-\frac{5}{\sqrt{37}}\cdot\frac{\sqrt{37}}{\sqrt{37}}\)
\(\sin\theta=\large-\frac{5\sqrt{37}}{37}\)
\(\cot\theta=\large-\frac{2\sqrt{3}}{5}\)
Use Trig ratios to evaluate expressions.
Example 2: Find the exact value of \(\cos\left(3\pi\right)+2\tan\left(\frac{3\pi}{4}\right)\).
\(1 + 2(-2)\)
\(-1\)
Guided Learning
Given a coordinate, find trig ratios.
Example 1: The coordinate \(\left(2\sqrt{3},-5\right)\) lies on the terminal side of \(\theta\). Find the exact values of \(\sin\theta\) and \(\cot\theta\).
Begin by sketching a picture.
Using Pythagorean theorem, we can find the hypotenuse of the right triangle with side lengths of \(5\) and \(2\sqrt{3}\).
\(\left(2\sqrt{3}\right)^2+5^2=c^2\)
\(12+25=c^2\)
\(c=\sqrt{37}\)
Once you know all three sides of the triangle, you can use them to write any trig ratio.
\(\sin\theta=\large-\frac{5}{\sqrt{37}}\)
\(\sin\theta=\large-\frac{5}{\sqrt{37}}\cdot\frac{\sqrt{37}}{\sqrt{37}}\)
\(\sin\theta=\large-\frac{5\sqrt{37}}{37}\)
\(\cot\theta=\large-\frac{2\sqrt{3}}{5}\)
Use Trig ratios to evaluate expressions.
Example 2: Find the exact value of \(\cos\left(3\pi\right)+2\tan\left(\frac{3\pi}{4}\right)\).
\(1 + 2(-2)\)
\(-1\)
Guided Learning