1 1 1 135 tan a we are going to use this method to

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1 1 1 0 135 tan - = - = = ° a We are going to use this method to find angles that are non acute, finding an acute reference angle, making a triangle and seeing which quadrant we are in to help with the signs. 2 2 2 1 135 cos - = - = = ° h a
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Let θ denote a nonacute angle that lies in a quadrant. The acute angle formed by the terminal side of θ and either the positive x -axis or the negative x -axis is called the reference angle for θ . Let's use this idea to find the 6 trig functions for 210° First draw a picture and label θ (We know that 210° will be in Quadrant III) Now drop a perpendicular line from the terminal side of the angle to the x -axis The reference angle will be the angle formed by the terminal side of the angle and the x -axis. Can you figure out it's measure? θ =210° 210°-180°=30° The reference angle is the amount past 180° of θ 30° Label the sides of the 30-60-90 triangle and include any negative signs depending on if x or y values are negative in the quadrant. 2 -1 3 -
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30° 210 ° 2 -1 3 - You will never put a negative on the hypotenuse. Sides of triangles are not negative but we put the negative sign there to get the signs correct on the trig functions. = ° 210 csc You should be thinking csc is the reciprocal of sin and sin is opposite over hypotenuse so csc is hypotenuse over opposite. 2 1 2 - = - = ° 210 tan 3 3 3 1 = - - = ° 210 cos 2 3 -
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Using this same triangle idea, if we are given a point on the terminal side of a triangle we can figure out the 6 trig functions of the angle. 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. First draw a picture (5, -12) Now drop a perpendicular line from the terminal side to the x -axis Label the sides of the triangle including any negatives. You know the two legs because they are the x and y values of the point 5 -12 Use the Pythagorean theorem to find the hypotenuse ( 29 ( 29 2 2 2 12 5 h = - + 13 = h 13
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