Presentation on theme: "Inverse Trigonometric Functions 4.7"— Presentation transcript:
1Inverse Trigonometric Functions 4.7 By: Ben O’Shasky, Luke Gause, and Tyler Gilbert
2Introduction to Inverse Trigonometric Functions The Inverse Trigonometric Functions are the inverse functions of the Trigonometric Functions. They are viewed as: sin−1, cos−1, tan−1 . You can evaluate them on your calculator or evaluate them without your calculator which we will be showing through our examples later on. You use the Inverse functions when you have the trigonometric ratio and you need to find the angle that has that trigonometric ratio.
3Unit Circle (0, 1) (1, 0) (-1, 0) (0, -1) 60º 45º Red= 30º angles Note:Cosine is negative in the II and III quadrant.Sine is negative in the III and IV quadrant.Tangent is negative in the II and IV quadrant.(0, 1)60º45ºRed= 30º anglesBlue= 45º anglesGreen= 60º anglesPurple= 90º angles30º(1, 0)(-1, 0)X= CosineY= SineY/X= Tangent(0, -1)
4Unit Triangles 45, 45, 90 Triangle 30, 60, 90 Triangle 45º √2 60º √3 1 30º45º21***Use the 45, 45, 90 triangle when you are dealing with an angle that is a 45º angle. Use the 30, 60, 90 triangle when you are dealing with a 30º or a 60º angle.Match the trigonometric ratio you are given (depending on if you are using sine, cosine, or tangent) to the angle (X) that you are trying to find.
5Evaluating Angles Using Inverse Y=sin-1 X X=sin YExample: sin-1 (1/√ 3) = XStep 1: Figure out which triangle to useThe fraction has numbers from the30, 60, 90 triangle. Since sine isOpposite side over Hypotenuse, theangle we are looking for must matchthis fraction.Step 2: The angle that has an opposite of 1 and has the hypotenuse of √3 is the 30º angle. Therefore the X= 30ºThe process is the same for cosine and tangent.Cosine is adjacent/hypotinuseTangent is opposite/adjacent√ 360º130º2
6Evaluating Inverse Using Calculator Steps on a Calculator2ndSin, cos, or tan (depending on what inverse you are using)Type in the trigonometric ratio.Make sure you have parentheses around ratio when needed.Example: cos(X) = 1/2cos-1 (1/2) = XFollow the calculator stepsCos-1 (1/2)Press “ENTER” to get the angle.In this example…. 60Unknown AngleTrigonometric Ratio
7Calculating a viewing angle In order to calculate a viewing angle you need to know two side lengths of a triangleEx.)106xArcsin(10/6) = xX =
8tan-1(1) cos-1(1/2) sin-1(1/√2) sin-1(1/2) tan-1(√3) Find the Exact Value Without using a calculator, than check your answers using a calculator.tan-1(1)cos-1(1/2)sin-1(1/√2)sin-1(1/2)tan-1(√3)
9Flash Card Answersπ /4 or 45ºπ /3 or 60ºπ /6 or 30º
10Multiple Choice Questions Find the exact value without a calculator for the ones without decimals and a calculator for the ones with decimals…Arcsin(√3/2)Arc sin(-1/√2)Arc cos(1/2)Arc sin( ½)Arctan (1/4)Arc cos( 3/5)Arcsin(2/3)Arctan(1/6)Arccos(5/6)Arctan(383/500)
11Multiple Choice A)60 degrees B) 45 degrees C) 30 degrees D) 90 degrees
12Multiple Choice Continued A) 50 degreesb) 75 degreesc) 30 degreesd) 105 degreesA) degreesB) degreesC) degreesD) degreesA) degreesB)72.541degreesC) degreesD) degrees
13Multiple Choice Continued 7. A) πB) .232πC).934πD)1.567πA) 9.462B)C)2.537D)49.321A)B)11.254C)89.352D)33.557A)50B)75.284C)60.302D)37.452