Secondary Math 3- Honors 7-4 Inverse Trigonometric Functions
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What You Will Learn How to restrict the domain of trigonometric functions so that the inverse can be constructed. How to use inverse functions to write the exact values of trigonometric expressions.
Review What is the definition of an inverse? formed when the independent variable (x) is exchanged with the dependent variable (y) in a given relation. The inverse of a function takes the output answer, performs an operation, and arrives back at the original function s value. Each point in an inverse relation is a point in the relation reflected across the line y = x (line through the origin). How do you know if the inverse of a function is also a function? If it passes the horizontal line test. If the function is a one-to-one function. What can you do to the original function if it is not oneto-one so its inverse is a function? Restrict the domain.
Inverse Trigonometric Functions The sine function does not pass the horizontal line test. We need to restrict the domain. Look for where the graph is always increasing or decreasing. The function is increasing on the interval,. By restricting the domain to this interval, it will allow its inverse to be constructed.
Inverse Trigonometric Functions The inverse of the sine function is denoted two ways: arcsin The x coordinates now represent the sine value and the y coordinates represent the unique angle (notice the scale on the graph) The domain is [-1, 1] and range is,
Inverse Trigonometric Functions The cosine function does not pass the horizontal line test. We need to restrict the domain. The function is decreasing on the interval 0,. By restricting the domain to this interval, it will allow its inverse to be constructed.
Inverse Trigonometric Functions The inverse of the cosine function is denoted two ways: arc The x coordinates now represent the cosine value and the y coordinates represent the unique angle (notice the scale on the graph) The domain is [-1, 1] and range is 0,
Inverse Trigonometric Functions The tangent function does not pass the horizontal line test. We need to restrict the domain. The function is increasing on the interval,. By restricting the domain to this interval, it will allow its inverse to be constructed.
Inverse Trigonometric Functions The inverse of the tangent function is denoted two ways: arc The x coordinates now represent the tangent value and the y coordinates represent the unique angle (notice the scale on the graph) The domain is (-, ) and range is,
Find the exact value of the expression without a calculator of This is really asking us to find the angle,, that is in the interval, whose sine value is. or The inverse sine function is restricted to the first and fourth quadrants of the unit circle. Sine is positive in the first quadrant, therefore, will be an angle from the first quadrant. 2 2 4
Examples find the exact value of the expression without a calculator 3 2 1 2
Find the exact value of the expression without a calculator of This is really asking us to find the angle,, that is in the interval 0, whose cosine value is. or cos The inverse cosine function is restricted to the first and second quadrants of the unit circle. Cosine is positive in the first quadrant, therefore, will be an angle from the first quadrant. 1 2 3
Examples find the exact value of the expression without a calculator 2 2 3 2
Find the exact value of the expression without a calculator of 1 This is really asking us to find the angle,, that is in the interval, whose tangent value is 1. 1 or tan 1 The inverse tangent function is restricted to the first and fourth quadrants of the unit circle. Tangent is positive in the first quadrant, therefore, will be an angle from the first quadrant. 1 4
Examples find the exact value of the expression without a calculator 3 3 3
Use a calculator to find the approximate value of each angle. Put your answer in degrees and round to the nearest thousandths. 0.378 67.79 11.67 85.102 1.45 Error the domain for inverse sine is [-1, 1] and 1.45 is not in that interval
The given point is on the terminal side of in standard position. Find the measure of. (-2, 4)
For inverse functions the following properties are true: ' Thus the following properties are true for trigonometric functions. for every x in the interval 1,1 x for every x in the interval, for every x in the interval 1,1 x for every x in the interval 0, for every x in the interval (-, ) x for every x in the interval,
Example Find the exact value for the expression. 0.3 0.3 0.3 x = 0.3 is in the interval [-1, 1]. Use the properties of inverse trigonometric functions '
Example Find the exact value for the expression. * + x = * + is not in the interval [0, π]. *, +, Evaluate the inside function Find the angle whose cosine value is,, - +
Example Find the exact value for the expression. x = is not in the interval (,. tan 0 0 Evaluate the inside function Find the angle whose cosine value is 0 0 0
Example Find the exact value for the expression. 100 Since 100 is between (-, ), Use the properties of inverse trigonometric functions ' 100 =100 2 3 x =, is not in the interval [, ]. Evaluate the inside function,,,,
Find the exact value of the expression tan -, Using the definition of the inverse cosine function, we can rewrite the inside function -,. The cosine value is positive, which means the angle we are looking for is in the first quadrant. It is not necessary to find the actual value of the angle because tan - is, asking us to find the tangent of the angle whose cosine is -., /012 34/2 and 4554/067 89:83716 Use the Pythagorean Theorem to find y. ; 13 5 144 12 tan -, -
Find the exact value of the expression cos = Using the definition of the inverse sine function, we can rewrite the inside function sin =. The sine value is negative, which means the angle we are looking for is in the fourth quadrant. It is not necessary to find the actual value of the angle because cos = is asking us to find the cosine of the angle whose sine is - =. cos 89:83716 >?54671@/7 Use the Pythagorean Theorem to find the adjacent side. ; 4 1 15 cos = - =
Find the exact value of the expression cos = * Using the definition of the inverse sine function, we can rewrite the inside function sin = *. The sine value is positive, which means the angle we are looking for is in the first quadrant.
Write sin as an algebraic expression if 0 < x 1. The expression is limited to the first quadrant because of the expression 0 < x 1. Using the definition of the inverse cosine function, we can rewrite the inside function as cos A. It is not necessary to find the actual value of the angle because sin is asking us to find the sine of the angle whose cosine is A. sin 4554/067 >?54671@/7 Use the Pythagorean Theorem to find the opposite side. b 1 sin AB 1