Given that the point 5 12 is on the terminal side of

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Given that the point (5, -12) is on the terminal side of an angle θ , find the exact value of each of the 6 trig functions. (5, -12) 5 -12 13 = = = = = = θ θ θ θ θ θ cot tan sec cos csc sin θ α We'll call the reference angle α . The trig functions of θ are the same as α except they possibly have a negative sign. Labeling the sides of triangles with negatives takes care of this problem. 12 13 o h - = 13 5 = h a 12 5 o a - = 12 13 - = o h 5 13 = a h 12 5 - = o a
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In quadrant I both the x and y values are positive so all trig functions will be positive + + α All trig functions positive In quadrant II x is negative and y is positive. _ + α We can see from this that any value that requires the adjacent side will then have a negative sign on it. Let's look at the signs of sine, cosine and tangent in the other quadrants. Reciprocal functions will have the same sign as the original since "flipping" a fraction over doesn't change its sign. sin is + cos is - tan is -
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_ _ α In quadrant IV, x is positive and y is negative . _ + α So any functions using opposite will be negative. Hypotenuse is always positive so if we have either adjacent or opposite with hypotenuse we'll get a negative. If we have both opposite and adjacent the negatives will cancel sin is - cos is + tan is - In quadrant III, x is negative and y is negative. sin is - cos is - tan is +
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All trig functions positive sin is + cos is - tan is - sin is - cos is + tan is - sin is - cos is - tan is + To help remember these sign we look at what trig functions are positive in each quadrant. A S T C Here is a mnemonic to help you remember. (start in Quad I and go counterclockwise) All Students Take Calculus
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What about quadrantal angles? We can take a point on the terminal side of quadrantal angles and use the x and y values as adjacent and opposite respectively. We use the x or y value that is not zero as the hypotenuse as well.
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