What is the analysis of the angles θ = 120° and θ = 135°?
Understand the Problem
The question presents two angles, θ = 120° and θ = 135°, likely asking for some analysis or calculation involving these angles, possibly in relation to trigonometric functions or geometric properties.
Answer
- For \( \theta = 120^\circ \): \( \sin(120^\circ) = \frac{\sqrt{3}}{2}, \cos(120^\circ) = -\frac{1}{2}, \tan(120^\circ) = -\sqrt{3} \) - For \( \theta = 135^\circ \): \( \sin(135^\circ) = \frac{\sqrt{2}}{2}, \cos(135^\circ) = -\frac{\sqrt{2}}{2}, \tan(135^\circ) = -1 \)
Answer for screen readers
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For ( \theta = 120^\circ ):
- ( \sin(120^\circ) = \frac{\sqrt{3}}{2} )
- ( \cos(120^\circ) = -\frac{1}{2} )
- ( \tan(120^\circ) = -\sqrt{3} )
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For ( \theta = 135^\circ ):
- ( \sin(135^\circ) = \frac{\sqrt{2}}{2} )
- ( \cos(135^\circ) = -\frac{\sqrt{2}}{2} )
- ( \tan(135^\circ) = -1 )
Steps to Solve
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Identify Trigonometric Functions for Each Angle
We will calculate the sine, cosine, and tangent for both angles.
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Calculate Trigonometric Values for θ = 120°
For the angle ( \theta = 120^\circ ):
- The sine is given by: $$ \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} $$
- The cosine is: $$ \cos(120^\circ) = -\cos(60^\circ) = -\frac{1}{2} $$
- The tangent is: $$ \tan(120^\circ) = \frac{\sin(120^\circ)}{\cos(120^\circ)} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} $$
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Calculate Trigonometric Values for θ = 135°
For the angle ( \theta = 135^\circ ):
- The sine is: $$ \sin(135^\circ) = \sin(180^\circ - 45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} $$
- The cosine is: $$ \cos(135^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2} $$
- The tangent is: $$ \tan(135^\circ) = \frac{\sin(135^\circ)}{\cos(135^\circ)} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1 $$
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For ( \theta = 120^\circ ):
- ( \sin(120^\circ) = \frac{\sqrt{3}}{2} )
- ( \cos(120^\circ) = -\frac{1}{2} )
- ( \tan(120^\circ) = -\sqrt{3} )
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For ( \theta = 135^\circ ):
- ( \sin(135^\circ) = \frac{\sqrt{2}}{2} )
- ( \cos(135^\circ) = -\frac{\sqrt{2}}{2} )
- ( \tan(135^\circ) = -1 )
More Information
The angles ( 120^\circ ) and ( 135^\circ ) are both found in the second quadrant of the unit circle, where sine values are positive, and cosine values are negative. This results in negative tangent values.
Tips
- Confusing the signs of the trigonometric functions based on the quadrant.
- Not using the appropriate reference angle when calculating the sine and cosine.
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