Inverse Trig Functions (part 2)

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Presentation transcript:

Inverse Trig Functions (part 2) I. Evaluating Inverse trig Functions with a Calculator. A) If the angles we are looking for aren’t on the Unit Circle (30°,45°,60° or 90°) we use a calculator. 1) By definition: we measure angles in radians. 2) Make sure your calculator is in radian mode. B) If we are looking for sin–1 x, cos–1 x or tan–1 x, then push that button on your calculator. 1) If it says arcsin, arccos, or arctan that means the same as trig–1 x so use the same buttons. C) You can also find the degree of these angles by doing the same math but use degree mode.

Inverse Trig Functions (part 2) II. Compositions of Trig Functions. A) We can do compositions of trig(arctrig x) where the two trig words are different. For example tan(arcsin 3/5). B) We need to use the Pythagorean Thm and our knowledge of what Quadrant in the Unit Circle sin, cos, & tan is + or –. 1) sin (+ I, II) , (– III, IV) [sin is the y values] sin = y/r 2) cos (+ I, IV) , (– II, III) [cos is the x values] cos = x/r 3) tan (+ I, III) , (– II, IV) [tan is the y/x values] tan = y/x C) Plug in the given values into Pythag. thm to get all 3 #s and determine what sign (positive or negative) the x & y values will be. Remember, the radius (r) is always positive. 1) Pythagorean thm: x2 + y2 = r2

Inverse Trig Functions (part 2) II. Compositions of Trig Functions. D) Now you can evaluate the composition. 1) Remember arctrig means to find the angle, so if we have trig(arctrig 3/5) we are saying trig(an angle) which will equal an number (not another angle). Examples: tan(arccos 2/3) cos = x/r so 22 + y2 = 32. therefore y = √5. Since tan = y/x and x is + we are in Quad I. Then tan(arccos 2/3) = √5/2 cos[arcsin (-3/5)] sin = y/r so x2 + (-3)2 = 52. therefore x = 4. Since sine is negative, we are in Quad IV (cos is + in Quad IV). And cos = x/r, then cos[arcsin (-3/5)] = 4/5

Inverse Trig Functions (part 2) II. Compositions of Trig Functions. E) How to determine if you are looking for an angle (θ) or a #. 1) arctrig (or trig–1) is an angle, anything else is a number 2) So trig(arctrig x)  trig(θ) = #. We find the number. 3) So arctrig(trig x)  arctrig(#) = θ. We find the angle. Examples: cos(arcsin x)  since arcsin x is an angle, this is really saying cos θ. So we want to get a number. We want the number x/r because cos θ = x/r. arcsin(tan x)  since tan x is a number y/x, this is really saying arcsin #. So we want the angle θ that gives us the value of sin y/r.