4.7a Trig Inverses.notebook September 18, 2014
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1 WARM UP Recall from Algebra 2 (or possibly see for the first time...): In order for a function to have an inverse that is also a function, it must be one to one, which means it must pass the horizontal line test. Consider the graph of y=sin(x), shown below. Answer the following questions to prepare for today's lesson: Does the graph of y=sin(x) pass the Horizontal Line Test? Based on the previous question, what can you conclude about the inverse of y=sin(x)? Restrict the domain so that the graph DOES pass the HLT. Oct 3 9:00 AM Sep 18 9:29 AM
2 Consider y = sin(x) This was just on the warm up... Is the function one to one? Is the inverse a function? Restrict the domain: Now is the function one to one? So is the inverse a function now? f(x)=sin(x) f(x)=sin 1 (x) Something else to remember from Algebra 2 about inverses... domain of f(x) => range of f 1 (x) range of f(x) => domain of f 1 (x) Oct 4 1:59 PM Graphing the inverse sine function... This is the restricted y = sin(x) graph. 2 1 x y x y 1 2 Oct 4 1:59 PM
3 Inverse Sine Inverse function is only defined where (where x is a ratio of sides or sine values) (where y is an angle in Quads or ) In plain language, the arcsine (or sin 1 ) of x is "the angle whose sine is x." Oct 4 2:06 PM WARNING: Oct 4 2:06 PM
4 Example 1: Evaluate 1) Rewrite: In other words, find the angle θ where the ratio of sides for sine is. (value of sine) 2) Determine the quadrant where this angle could exist: Normally, sin(θ) is positive in Quad I & II, but we can only look at angles in Quad I because the inverse sine is only defined in QI & IV. Think about your unit circle!!! 3) Check with calculator: Input Calculator will NOT give exact values in terms of π! Oct 4 4:01 PM Try #9(a), p. 327 (HW) Find sin Oct 4 4:01 PM
5 Consider y = cos(x) Is the function one to one? Is the inverse a function? Restrict the domain: Now is the function one to one? So is the inverse a function now? f(x)=cos(x) f(x)=cos 1 (x) Oct 4 1:59 PM Inverse Cosine y = arccos (x) or y = cos 1 (x) iff x = cos y Inverse function is only defined where (where x is a ratio of sides or cosine values) (where y is an angle in Quads or ) In plain language, the arccosine (or cos 1 ) of x is "the angle whose cosine is x." Oct 4 2:06 PM
6 WARNING: y = cos 1 (x) = 1 cos(x) Oct 4 2:06 PM Example 2: Evaluate 1) Rewrite: In other words, find the angle y where the ratio of sides for cosine is. (value of cosine) 2) Determine the quadrant where this angle could exist: Normally, cos(θ) is negative in Quad II & III, but we can only look at angles in Quad II because the inverse cosine is only defined in Quad I & II. Think about your unit circle!!! 3) Check with calculator: Input Calculator will NOT give exact values in terms of π! Oct 4 4:01 PM
7 Another example: Find cos Oct 4 4:01 PM Consider y = tan(x) Is the function one to one? Is the inverse a function? Restrict the domain: Now is the function one to one? So now is the inverse a function? f(x)=tan(x) f(x)=tan 1 (x) Oct 4 1:59 PM
8 Inverse Tangent y = arctan (x) or y = tan 1 (x) iff x = tan y Inverse function is only defined where (where x is a ratio of sides or cosine values) (where y is an angle in Quads or ) In plain language, the arctangent (or tan 1 ) of x is "the angle whose tangent is x." Oct 4 2:06 PM WARNING: y = tan 1 (x) = 1 tan(x) Oct 4 2:06 PM
9 Small shortcut to find tangent while looking at the unit circle... tan = 6 *Remember tan = sin/cos tan = numerator of sin numerator of cos Sep 17 8:14 PM Example 3: Evaluate 1) Rewrite: In other words, find the angle y where the ratio of sides for tangent is. (value of tangent) Check with calculator: Input In this case, calculator did give the exact value. Oct 4 4:01 PM
10 Another example: Find tan Oct 4 4:01 PM We can use inverse trig functions to solve for angles in a right triangle. x Find θ in terms of x for all 3 inverse trig functions. 4 Note: The ratio of sides must always be between 1 and 1 for sin 1 (x) and cos 1 (x). Calculator will produce an error otherwise. Oct 4 4:34 PM
11 Summary Function Range Quadrants Oct 3 9:00 AM HOMEWORK...on a clean sheet of paper, due tomorrow 4.7a (p. 327): 1 9 (odd), 13, 14, 15 27(odd) Oct 6 10:14 AM
12 Inverse trig ratios allow us to solve for ANGLE values that correspond to a given ratio of sides. EX: Find arcsin(.56) using a calculator. In radian mode: In degree mode: Find approximate angle values rounding to 3 decimal places. Oct 4 4:31 PM
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