y = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions
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1 MATH 7 Right Triangle Trig Dr. Neal, WKU Previously, we have seen the right triangle formulas x = r cos and y = rsin where the hypotenuse r comes from the radius of a circle, and x is adjacent to and y is opposite of. But these formulas hold in any right triangle with hypotenuse, even outside the context of a circle: (radius) r y = rsin (opp) (hyp) y = sin (opp) x = r cos (adj) x = cos (adj) Solving for cos and sin, we have cos = x = Adjacent Hypotenuse sin = y = Opposite Hypotenuse The Other Trig Functions We define four other trigonometric functions the tangent, the cotangent, the secant, and the cosecant as follows: Tangent: tan = y x = Opposite Adjacent Secant: sec = cos = x = Hypotenuse Adjacent Cotangent: cot = tan = x y = Adjacent Opposite Cosecant: csc = sin = y = Hypotenuse Opposite Finding the Trig Function Values and Angles Given Two Sides Given any two sides of a right triangle, we can use the Pythagorean Theorem to find the third side. Then we can compute the six trig function values of the acute angles in the triangle by using the above ratios. We also can use a calculator to approximate the measure of the angles.
2 Example. In the triangle below: (a) Find the remaining side. (b) Find the six trig. function values of the acute angles. (c) Find the approximate degree measures of the acute angles. " c Solution. By the Pythagorean Theorem, c 2 = ; so c = = 4225 = With respect to angle, adj = opp = hyp = cos = adj hyp = sin = opp hyp = tan = opp adj = cot = tan = sec = csc = cos = sin = With respect to angle, adj = opp = hyp = cos = adj hyp = sin = opp hyp = tan = opp adj = cot = tan = sec = cos = csc = sin = To find the measures of and, put your calculator in degree mode and evaluate COS (/) and COS (/) The COS (inverse cosine) button is 2ND COS. Then, = cos " # & % ( ) 30.5º and = cos " # & % ( ) 59.49º $ ' $ '
3 Example 2. In the triangle below: (a) Find the remaining side. (b) Find the six trig. function values of the acute angles. (c) Find the approximate degree measures of the acute angles. d 57 " Solution. Because 2 + d 2 = 57 2 ; we have 32 d = = 7424 = " With respect to angle, adj = opp = 32 hyp = 57 cos = adj hyp = 57 sin = opp hyp = tan = opp adj = 32 cot = tan = 32 sec = cos = 57 csc = sin = With respect to angle, adj = 32 opp = hyp = 57 cos = adj hyp = sin = opp hyp = 57 tan = opp adj = 32 cot = tan = 32 sec = cos = csc = sin = 57 The measures of and are = cos " # & % ( ) º and = cos " # 32& % ( ) 32.78º. $ 57' $ 57'
4 Finding the Remaining Sides Given a Side and an Angle Given a side and one acute angle in a right triangle, then the other angle is simply the complement of the first (i.e., they add up to 90º). The other two sides can be found by using the appropriate right triangle ratio. Example 3. Find the remaining pieces of the following right triangles using only the given labeled information on each: 2 b w 8 55º 20 c (i) 40º a (ii) (iii) a 30º Solution. (i) First, = 50º, which is the complement of the given 40º angle. But is not needed to find the other sides. We need to apply the appropriate formula below: cos = Adj Hyp sin = Opp Hyp tan = Opp Adj Part (i) With respect to = 40º, we have: adj = a opp = b hyp = 2 So, cos 40º = Adj Hyp = a 2 and sin 40º = Opp Hyp = b 2 a = 2cos 40º 9.9 (via calculator) b = 2sin 40º 7.7 (via calculator) Part (ii) With respect to = 55º, we have: adj = 8 opp = w hyp = So, cos 55º = Adj Hyp = 8 = 8 cos55º (via calculator) and tan 55º = Opp Adj = w 8 w = 8tan 55º.425 (via calculator)
5 Part (iii) With respect to = 30º, we have: adj = a opp = 20 hyp = c Dr. Neal, WKU So, sin 30º = Opp Hyp = 20 c c = 20 sin30º = 20 2 = 40 and tan 30º = Opp Adj = 20 a a = 20 tan 30º = 20 3 = 20 3 Right Triangle Trig on the xy Plane Any set of ( x, y ) coordinates determines an angle by considering the segment from (0, 0) to ( x, y ). The trig function values can be quickly found by using cos = x sin = y tan = y x where we always take = + x 2 + y 2. Example 4. Find the six trig function values of the positive angle determined by the following points. Then find the approximate measure of the angle. (i) ( 9, 40) (ii) ( 80, 39) (iii) (88, 05) Solution. In each case, we first need to find the hypotenuse. Then we apply the right triangle trig formulas to find the six trig function values. Finally, we'll use tan (y/x) to find the angle. In each case though, we will have to add either 80º or 360º to adjust for the correct quadrant of. Part (i): First, x = 9, y = 40, and = = 68 = 4. Then, cos = "9 4 sin = 40 4 tan = 40 "9 cot = "9 40 In degree mode: tan (40/ 9) 77.32º which is in the 4th Quadrant. But ( 9, 40) is in the 2nd Quadrant, so add 80º to adjust: 77.32º + 80º 02.68º. sec = 4 " º ( 9, 40) csc = º
6 Part (ii): First, x = 80, y = 39, and = = 792 = 89. So, cos = " sin = " tan = cot = sec = " csc = " In degree mode: tan ( 39/ 80) 25.99º which is in the st Quadrant. But the given point ( 80, 39) is in the 3rd Quadrant, so add 80º to adjust º Then 25.99º + 80º º º ( 80, 39) Part (iii): We have x = 88, y = 05, and = = 8769 = 37. Thus, cos = sin = " tan = " cot = " sec = csc = " In degree mode: tan ( 05/88) 50º which is in the 4th Quadrant as is (88, 05). But to get a positive angle, add 360º. So we actually have 30º. Finding Other Trig Function Values of θ Given one trig function value of an angle and its quadrant, we can make an (x, y, ) system to find the other trig function values. Always keep the hypotenuse positive, and use the appropriate +/ signs for x and y depending on the quadrant. Example 5. Find the remaining trig function values of the described angle. (i) sec = " 7 5 and is in III (ii) csc = 9 4 and is in II (iii) cot = " 5 2 and is in IV Solution. (i) For sec = " 7 "5, then cos = 5 7 = x. And y is also negative, because is in Quadrant III. Then y = 2 x 2 = 49 (5) 2 = 24 = 2 6. The other four trig function values are:
7 sin = y = " csc = y = " tan = y x = cot = x y = 5 III (ii) For csc = 9 4, then sin = 4 9 = y. But x is negative, because is in Quadrant II. So x = 2 y 2 = 8 6 =. The other four trig function values are: cos = x = " 9 tan = y x = " 4 sec = x = " 9 cot = x y = " 4 4 II 9 (iii) cot = " 5 2 tan = "2 5 = y x 5 ( is in IV, so x is pos. and y is neg.) = (2) 2 = Quad IV cos = x = 5 3 sin = y = "2 3 sec = x = 3 5 csc = y = " 3 2
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