Problem: Calculate the Sine, Cosine, and Tangent Values of Specific Angles on the Unit Circle

Let’s determine the sine, cosine, and tangent values for the angle θ = 225° on the unit circle.

First, convert the angle to radians:

$ θ = 225° = \frac{225π}{180} = \frac{5π}{4} radians $

Using the properties of the unit circle, we know:

$ \cos(\frac{5π}{4}) = -\frac{\sqrt{2}}{2} $

$ \sin(\frac{5π}{4}) = -\frac{\sqrt{2}}{2} $

$ \tan(\frac{5π}{4}) = \frac{\sin(\frac{5π}{4})}{\cos(\frac{5π}{4})} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 $

Thus, the sine, cosine, and tangent values for θ = 225° are:

$ \sin(225°) = -\frac{\sqrt{2}}{2} $

$ \cos(225°) = -\frac{\sqrt{2}}{2} $

$ \tan(225°) = 1 $

Consider the angle θ = 300° on the unit circle. To find the sine, cosine, and tangent values, convert the angle to radians:

$ θ = 300° = frac{300π}{180} = frac{5π}{3} radians $

From the unit circle:

$ cos(frac{5π}{3}) = frac{1}{2} $

$ sin(frac{5π}{3}) = -frac{sqrt{3}}{2} $

$ an(frac{5π}{3}) = frac{sin(frac{5π}{3})}{cos(frac{5π}{3})} = frac{-frac{sqrt{3}}{2}}{frac{1}{2}} = -sqrt{3} $

Therefore, the sine, cosine, and tangent values for θ = 300° are:

$ sin(300°) = -frac{sqrt{3}}{2} $

$ cos(300°) = frac{1}{2} $

$ an(300°) = -sqrt{3} $

For the angle θ = 135°:

$ θ = 135° = frac{135π}{180} = frac{3π}{4} radians $

Using the unit circle properties:

$ cos(frac{3π}{4}) = -frac{sqrt{2}}{2} $

$ sin(frac{3π}{4}) = frac{sqrt{2}}{2} $

$ an(frac{3π}{4}) = -1 $

Thus, the values are:

$ sin(135°) = frac{sqrt{2}}{2} $

$ cos(135°) = -frac{sqrt{2}}{2} $

$ an(135°) = -1 $

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