4.7 Inverse Trigonometric Functions

Transcription

1 47 Inverse Trigonometric Functions Section 47 Notes Page 1 From our tables in a previous section we know that sin 0 = In inverse trig functions we put in the value and get an angle: 1 We put in an angle and get a value as a result 1 sin 1 = 0 So here we put in the value of one half and got 0 degrees as a result We are not allowed to put any number into our inverse trig functions There are restrictions on the domain that are given in the following table: Domain Range y = sin x x 1 y y = cos x x 1 0 y y = tan x < x < < y < NOTE: in some textbooks, the inverse functions are written differently, for example instead of y = sin x, some textbooks may write this as y = arcsin x So instead of the symbol, it is replaced by the word arc These two mean exactly the same thing So y = arccos x would mean the same as y = cos x, etc EXAMPLE: Find the sin 1 What this is really asking is: find an angle between and that has a value of If you look on your table of values, go to the sine column and go down until you see the value 0 degrees, which is the answer This corresponds to an angle of EXAMPLE: Find the cos 1 What this is really asking is: find an angle between 0 and that has a value of If you look on your table of values, go to the cosine column and go down until you see the value angle of 45 degrees, which is the answer This also corresponds to an EXAMPLE: Find the tan 1 What this is really asking is: find an angle between and that has a value of If you look on your table of values, go to the tangent column and go down until you see the value angle of 0 degrees, which is the answer This also corresponds to an

2 EXAMPLE: Use a calculator to find, cos 0 7 Assume your angle is in radians Section 47 Notes Page if possible Round your answer to two decimal places We need to make sure our calculator is in radian mode before we proceed The inverse cosine is above the cosine key on your calculator You will probably need to use your second key in order to get the inverse cosine Your answer should be 08 If you got an error, try entering 07 first and then get the inverse EXAMPLE: Use a calculator to find, sin 1 ( ) Assume your angle is in radians if possible Round your answer to two decimal places If you try putting this in your calculator you will get an error This is because 1 is not in our domain Recall that the domain for the inverse sine function is x 1 This means we can only put in numbers between -1 and 1 So the answer is no solution Inverses and canceling If we take cos 1 (cos x) what will we get? Well, the inverse cosine and cosine will cancel and that will leave us with just x However there are some restrictions on what x can be as listed below: cos 1 (cos x) = x if 0 x cos(cos 1 x ) = x if x 1 sin 1 (sin x ) = x if x sin(sin 1 x ) = x if x 1 tan 1 (tan x ) = x if < x < tan(tan 1 x) = x if < x < EXAMPLE: Find the exact value if possible: tan(tan 1 5) According to our restrictions above, x can be any number, so tan(tan 1 5) = 5 EXAMPLE: Find the exact value if possible: sin sin The fraction 98/99 is 9898, and this is less than 1, so sin sin = 99 EXAMPLE: Find the exact value if possible: cos( cos ) The square root changed into a decimal is 141, which is bigger than 1, so the answer is no solution since 141 is not in our domain

3 cos 1 cos Since is in the domain 0 y, then our properties tell us cos 1 cos = Section 47 Notes Page sin 1 sin Be careful on this one The answer is not This is because is not in y So we need to evaluate inside the parenthesis first, and then take the inverse So we can use either a unit circle or reference angles to find sin = So now our problem becomes sin 1 interval y that has a y value of So sin 1 sin = We want to find an angle in the Since sine is positive in the first quadrant, our answer is 5 tan 1 tan 5 5 The answer is not This is because is not in < y < So we need to evaluate inside the parenthesis first, and then take the inverse So we can use either a unit circle or reference angles to find 5 tan = So now our problem becomes tan ( ) We want to find an angle in the interval y that has a y value of Since tangent is negative in the fourth quadrant, our answer is So 5 tan 1 tan = EXAMPLE: Use a sketch to find the exact value: sin cos 5 These problems involve drawing a triangle and labeling the sides like we did in a previous section The inverse trig function will tell you where to draw the triangle In our example there is an inverse cosine The inverse cosine s range will tell us where we can draw the triangle From the last section, the range for the inverse cosine is 0 y This corresponds to the first and second quadrant Since the fraction is positive, the 5 only quadrant the triangle can be drawn in is the first quadrant We know that the adjacent side is and the hypotenuse is 5 The Pythagorean Theorem will give us the opposite side, which is 4

4 Section 47 Notes Page The sine on the outside of our problem tells us how to write our answer From our drawing, sine is 4 over 5, so we write our θ answer as: 4 sin cos = 5 5 EXAMPLE: Use a sketch to find the exact value: cos sin 1 The inverse trig function will tell you where to draw the triangle, and in this case we have an inverse sine The inverse sine s range will tell us where we can draw the triangle From the last section, the range for the inverse sine is y This corresponds to the first and fourth quadrant Since the fraction inside the inverse is negative, the only quadrant the triangle can be drawn in is the fourth quadrant We know that the opposite side is -1 and the hypotenuse is The Pythagorean Theorem will give us the adjacent side, which is 5 The cosine on the outside of our problem tells us how to write our answer From our drawing, cosine is 5 over, so we write 5 our answer as: θ -1 cos sin 1 = 5 = 0 EXAMPLE: Use a sketch to find the exact value: tan cos 4 The inverse trig function will tell you where to draw the triangle, and in our case there is an inverse cosine The inverse cosine s range will tell us where we can draw the triangle From the last section, the range for the inverse cosine is 0 y This corresponds to the first and second quadrant Since the fraction inside the inverse is negative, the only quadrant the triangle can be drawn in is the second quadrant We know that the adjacent side is - and the hypotenuse is 4 The Pythagorean Theorem will give us the opposite side: 7 4 The tangent on the outside of our problem tells us how to write our 7 answer From our drawing, tangent is 7 over -, so: θ - tan cos = 4 7

5 Section 47 Notes Page 5 csc tan 1 The inverse trig function will tell you where to draw the triangle, and in this case we have an inverse tangent The inverse tangent s range will tell us where we can draw the triangle From the last section, the range for the inverse tangent is y This corresponds to the first and fourth quadrant Since the fraction inside the inverse is negative, the only quadrant the triangle can be drawn in is the fourth quadrant We know that the opposite side is -1 and the adjacent is The Pythagorean Theorem will give us the adjacent side: 10 The cosecant on the outside of our problem tells us how to write θ our answer From our drawing, cosecant is 10 over -1, so we -1 write your answer as: csc tan = = 10 Assume that u is positive and that the given inverse trigonometric function is defined for the expression in u EXAMPLE: Use right triangles to write in algebraic form: sec( tan 4u) These problems involve drawing a triangle and labeling the sides with algebraic expressions For all these problems we will assume that x is positive and the triangle should be drawn in the first quadrant We can 4u rewrite our problem as: sec tan We know that the adjacent side is 1 and the opposite side is 4u We can 1 use the Pythagorean theorem to find the hypotenuse: c = ( 4u) + (1) So we have c = u u +1 4u The secant on the outside of our problem tells us how to write θ our answer From our drawing, secant is 1u + 1 over 1 so we 1 write our answer as: sec ( tan 4u ) = 1u + 1

6 EXAMPLE: Solve the triangle: Section 47 Notes Page B I will let side BC be x Now we can use the Pythagorean Theorem to find find it: ( ) + x = 5 Solving this will give us: 9 + x = 45, so 5 x = To find m A, we can set up the following trig equation: cos A = So we have cos A = We need to take the inverse 5 cosine to get our answer So A = cos = 44 To find m B A cm C we will subtract 90 degrees and 44 degrees from 180 degrees We will get: m B = = 5 Now our triangle is solved 1 cos x = In order to solve this we need to isolate the inverse trig function, so we will first divide both sides by : cos x = Now in order to cancel out the inverse cosine we need to take the cosine of both sides: cos( cos x ) = cos On the left side the cosine and inverse cosine will cancel x = cos = 0 To get the value of cos, we need to look on the table, so x = 0 5sin x = sin x We need to get all the inverse trig functions on one side of the equation, so we will first subtract the inverse sine from both sides: sin 1 x = Now add the pi to both sides 1 sin x = Divide both sides by sin x = Now take the sine of both sides sin ( sin ) = sin x The sine and inverse sine cancel from the left side x = sin This is the same as negative 0 degrees From our table, sin 0 is x = The answer is negative because negative 0 degrees is in the 4 th quadrant, so sine is negative