How to Solve Trig Ratios of General Angles? (+FREE Worksheet!)
Trig Ratios of General Angles – Example 1:
Find the trigonometric function: \(cos\) \(120^\circ\)
Solve Trig Ratios of General Angles: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Unit-circle cosine
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Try one before moving on
Solve Trig Ratios of General Angles: pop-up practice
Solution:
\(cos\) \(120^{\circ}\)
Use the following property: \(cos\)\((x)=\) \(sin\)\((90^{\circ}-x)\)
\(cos\) \(120^{\circ} =\) \(sin\) \(( 90^{\circ} -120^{\circ})=\) \(sin ( -30^{\circ}) \)
Now use the following property: \(sin (-x)\)\(=- sin (x)\)
Then: \(sin ( -30^{\circ})=-sin (30^{\circ}\))\(=-\frac{1}{2 }\)
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Trig Ratios of General Angles – Example 2:
Find the trigonometric function: \(sin\) \(135^\circ\)
Solution:
Use the following property: \(sin\)\((x)=\) \(cos\)\((90^\circ-x)\)
\(sin\) \(135^\circ=\) \(cos\)\((90^\circ-135^\circ)=\) \(cos\)\((-45^\circ)\)
Now use the following property: \(cos\)\((-x)=cos x\)
Then: \(cos\)\((-45^\circ)=\) \(cos\)\((45^\circ)=\frac{\sqrt{2}}{2 }\)
Trig Ratios of General Angles – Example 3:
Find the trigonometric function: \(sin\) \(-120^\circ\)
Solution:
Use the following property: \(sin\)\((-x)=-\) \(sin\)\((x)\)
\(sin\)\(-120^\circ=-\) \(sin\) \(120^\circ\), \(sin\)\(120^\circ=\frac{\sqrt{3}}{2}\)
Then: \(sin\)\(-120^\circ=-\frac{\sqrt{3}}{2}\)
Trig Ratios of General Angles – Example 4:
Find the trigonometric function: \(cos\) \(150^\circ\)
Solution:
\(cos\) \(150^{\circ}\)
Use the following property: \(cos\)\((x)=\) \(sin\)\((90^{\circ}-x)\)
\(cos\) \(150^{\circ} =\) \(sin\) \(( 90^{\circ} -150^{\circ})=\) \(sin ( -60^{\circ}) \)
Now use the following property: \(sin (-x)\)\(=- sin (x)\)
Then: \(sin ( -60^{\circ})=-sin (60^{\circ}\))\(= -\frac{\sqrt{3}}{2}\)
Exercises
Use a calculator to find each. Round your answers to the nearest ten–thousandth.
- \(\color{blue}{sin \ – 120^\circ}\)
- \(\color{blue}{sin \ 150^\circ}\)
- \(\color{blue}{cos \ 315^\circ}\)
- \(\color{blue}{cos \ 180^\circ}\)
- \(\color{blue}{sin \ 120^\circ}\)
- \(\color{blue}{sin \ – 330^\circ }\)
Download Trig Ratios of General Angles Worksheet
- \(\color{blue}{-\frac{\sqrt{3}}{2}}\)
- \(\color{blue}{\frac{1}{2}}\)
- \(\color{blue}{\frac{\sqrt{2}}{2}}\)
- \(\color{blue}{-1}\)
- \(\color{blue}{\frac{\sqrt{3}}{2}}\)
- \(\color{blue}{\frac{1}{2}}\)
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