How to Solve Trig Ratios of General Angles? (+FREE Worksheet!)
How to solve trig ratios of general angles, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
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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