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Notes March 8 Chapter 6-4 Objective: Inverse Trig Functions Name_____________ Be able to evaluate inverse trigonometric functions Lesson: Inverse functions interchange the coordinates of the ordered pairs of the function so the domain, x, becomes the range, y, and the range, y becomes the domain, x. Comparison Table: sin(x) Graph: sin-1(x) or arcsin(x) Graph: Domain ____________ Domain __________ Range________ Range __________ (____, ____) (____,_____) (____, ____) (____, ____) __________ __________ (failed vertical line test) Sin(x) = __________ x = __________ Sin(x) is the value of sine of an angle x is .3393 x is all the ______ whose sine is .3393 The Inverse Functions are written: Ex. sin(x) = y A = cos(b) tan(θ) = z __________ or __________ __________ or __________ __________ or __________ Ex. Find all positive values of x for which cosx = ½ (look on your unit circle) x = cos-1 (1/2) so x = __________ and __________ so between 0 and 360, we have: ______ and ______ Find all positive values of x for which sin x = (1/√2) between 0 and 360 sinx = √2/2 Ex. Evaluate each expression. Assume that all angles are in quad 1 cos(arccos (1/2) = cos(arccsc(5/3)) = Notes March 8 Chapter 6-4 Inverse Trig Functions sec(arctan (7/13)) = sin(arcsin(13/14) = cot(tan(5/10) = Ex. Verify each equation. arccos √3/2 + arcsin √3/2 = arctan 1 + arccot 1 tan-1 ¾ + tan-1 5/12 = tan-1 56/33 put in calculator arcsin 3/5 + arcos 15/17 = arctan77/36 put in calculator Homework: pg. 331 16 – 44 even Name_____________ Notes March 8 Chapter 6-4 Inverse Trig Functions Name_____________ What I will be doing and the student will be filling in Lesson Plan #21 Inverse Trig Functions Chapter 6-4 Objective: Be able to evaluate inverse trigonometric functions Introduction: Remember when you talked about inverse functions before? For instance if y = 2x+3, then the inverse of y = (x-3)/2. Or for y = x2 the inverse was y = √𝑥? It’s like putting on your socks before shoes, and the inverse is taking of the shoes first and then the socks. We have already briefly discussed inverses of trig functions, with arc-sine, arc-cosine, and arctangent. These are different than cosecant, secant, and cotangent because the inverse functions are not really functions because they have multiple y values for the same x values. Lesson: Recall inverse functions interchange the coordinates of the ordered pairs of the function so the domain becomes the range and the range becomes the domain. Do Table: Sin(x) Domain (all reals) Range(-1<y<1) (x, sin(x)) (90, 1) Function sin-1(x) or arcsin(x) Domain (-1<y<1) Range (all reals) (sin(x), x) (1, 90) not a function (failed vertical line Sin(x) = (0.3393) The value of sine of an angle x is .3393 x = sin-1(x) or arcsin(x) cos-1 (A) = b tan-1 z = θ x = sin-1(0.3393) x is all the angles whose sine is .3393 test) Sin(x) = y A = cos(b) Tan(θ) = z Show pictures ex. Find all positive values of x for which cosx = ½ look on your unit circle x = cos-1 (1/2) so x = 60, 300, 420….. and π/3, 5 π/3, 7 π/3…… so between 0 and 360, we have 60 and 300 Find all positive values of x for which sin x = (1/√2) between 0 and 360 Notes March 8 Chapter 6-4 Inverse Trig Functions Sinx = √2/2 so 45, 135 Ex. Evaluate each expression. Assume that all angles are in quad 1 cos(arccos (1/2) = ½ cos(arccsc(5/3)) = 4/5 sec(arctan (7/13)) = √365/7 On their own: sin(arcsin(13/14) = 13/14 cot(tan(5/10) = 10/5 Ex. Verify each equation. arccos √3/2 + arcsin √3/2 = arctan 1 + arccot 1 30 + 60 = 45 +45 -1 tan ¾ + tan-1 5/12 = tan-1 56/33 put in calculator 36.9 + 22.6 = 59.5 arcsin 3/5 + arcos 15/17 = arctan77/36 put in calculator 36.9 + 28.1 = 65 Homework: pg. 331 16 – 44 evens Name_____________