![]() ![]() ![]() To help remember these sign we look at what trig functions are positive in each quadrant. So any functions using opposite will be negative. If we have both opposite and adjacent the negatives will cancel _ _ sin is -cos is -tan is + In quadrant IV, x is positive and y is negative. Hypotenuse is always positive so if we have either adjacent or opposite with hypotenuse we'll get a negative. + _ In quadrant III, x is negative and y is negative. We can see from this that any value that requires the adjacent side will then have a negative sign on it. + sin is +cos is -tan is - In quadrant II x is negative and y is positive. Reciprocal functions will have the same sign as the original since "flipping" a fraction over doesn't change its sign. + _ All trig functions positive In quadrant I both the x and y values are positive so all trig functions will be positive + Let's look at the signs of sine, cosine and tangent in the other quadrants. Labeling the sides of triangles with negatives takes care of this problem. The trig functions of are the same as except they possibly have a negative sign. Given that the point (5, -12) is on the terminal side of an angle , find the exact value of each of the 6 trig functions. You know the two legs because they are the x and y values of the point Use the Pythagorean theorem to find the hypotenuse ![]() ![]() First draw a picture 5 Now drop a perpendicular line from the terminal side to the x-axis -12 13 (5, -12) Label the sides of the triangle including any negatives. 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. 210° 30° -1 2 You should be thinking csc is the reciprocal of sin and sin is opposite over hypotenuse so csc is hypotenuse over opposite. Sides of triangles are not negative but we put the negative sign there to get the signs correct on the trig functions. You will never put a negative on the hypotenuse. The reference angle is the amount past 180° of Can you figure out it's measure? 30° -1 2 210°-180°=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. Let's use this idea to find the 6 trig functions for 210° First draw a picture and label (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 =210° The reference angle will be the angle formed by the terminal side of the angle and the x-axis. The acute angle formed by the terminal side of and either the positive x-axis or the negative x-axis is called the reference angle for . Let denote a nonacute angle that lies in a quadrant. =135° 1 45° -1 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. Notice the -1 instead of 1 since the terminal side of the angle is in quadrant II where x values are negative. Putting the negative on the 1 will take care of this problem. x values are negative in quadrant II so put a negative on the 1 =135° 1 45° -1 Now we are ready to find the 6 trig functions of 135° The values of the trig functions of angles and their reference angles are the same except possibly they may differ by a negative sign. When you label the sides if you include any signs on them thinking of x & y in that quadrant, it will keep the signs straight on the trig functions. (The sides might be multiples of these lengths but looking as a ratio that won't matter so will work) This is a Quadrant II angle. Let's label the sides of the triangle according to a 45-45-90 triangle. What is the measure of this reference angle? 180°- 135° = 45° Let's make a right triangle by drawing a line perpendicular to the x-axis joining the terminal side of the angle and the x-axis. This is an acute angle and is called the reference angle. HINT: Since it is 360° all the way around a circle, half way around (a straight line) is 180° =135° referenceangle If is 135°, we can find the angle formed by the negative x-axis and the terminal side of the angle. To do this we'll place angles on a rectangular coordinate system with the initial side on the positive x-axis. Now we will see how we can find the trig function values of any angle. Our method of using right triangles only works for acute angles. ![]()
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