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Warm Up 1) Solve: sin 𝜃 = 1 2 for 0° ≤ 𝜃 ≤ 360° 30 °, 150 ° 2) Solve: cos 𝜃 = − 3) If 𝑠𝑖𝑛𝜃 = 4) If cos𝜃 = 7 − 11 1 − 2 2 2 𝑓𝑜𝑟 0 ≤ 𝜃 ≤ 2𝜋 𝟑𝝅 𝟓𝝅 , 𝟒 𝟒 find 𝑐𝑜𝑡𝜃 in exact form. Quad IV 𝟕𝟐 𝟔 𝟐 − =− 𝟕 𝟕 find 𝑐𝑠𝑐𝜃 in exact form. Quad II 𝟐 𝟑 𝟑 5) What is the 𝑐𝑜𝑡𝜃 ÷ 𝑐𝑠𝑐𝜃 ÷ 𝑠𝑒𝑐𝜃? 𝒄𝒐𝒔𝟐 𝜽 Section 7-6 The Inverse Trigonometric Functions Objective: To find values of the inverse trigonometric functions Trig FUNCTIONS Sine, cosine and tangent are all functions Are they all one-to-one functions? Domain: {x | x 2 n } Does the graph have an inverse? No! Notice how the axes are scaled! Domain: {x | x How can you restrict the domain to make the graph one-to-one? 2 n } 𝑓 𝑥 = 𝑇𝑎𝑛𝑥 Tan x has an inverse. Notice T is capitalized Notice how the axes are scaled! Domain: {x | x } 2 2 Notice how the axes are scaled! Notice how the axes are scaled! How can you restrict the domain to make the graph one-to-one? 𝜋 𝜋 𝑥 − ≤𝑥≤ 2 2 Restrict domain to: F(x)= Sin x f(x)= sin x y 3 2 1 x -3π/2 -π -π/2 π/2 -1 -2 -3 π 3π/2 Inverse function is Sin-1 x y π/2 x -2 1 -1 2 -π/2 Notice how the axes are scaled! 𝑓 𝑥 = 𝑐𝑜𝑠𝑥 How can you restrict the domain to make the graph one-to-one? Restrict domain to 0 <x < y F(x)= Cos x f(x)= cos x 3 2 1 x -3π/2 -π -π/2 π/2 -1 -2 -3 π 3π/2 𝑭−𝟏 𝒙 = 𝑪𝒐𝒔−𝟏 𝒙 y 3π/2 π π/2 x -2 -1 1 -π/2 -π -3π/2 2 Sin { } 2 2 Q1 & Q4 OR Sin { 90 90 } 𝐶𝑜𝑠𝑥 𝑇𝑎𝑛𝑥 Tan { OR Cos { 0 } } 2 2 Q1 & Q4 Tan { 90 90 } 𝑆𝑖𝑛𝑥 OR Q1 & Q2 Cos { 0 180} Inverse Trig Functions Remember, finding the inverse is finding an angle! 𝑠𝑖𝑛 −1 1 = 30° 2 Because: sin 30 ° = 1 2 Example 1 Find Tan-1 2 in radians with a calculator. First make sure your calculator is in the correct mode. Example 2 Find Tan-1 (-1) without a calculator. Find Tan1 1 without a calculator. "The number angle whose tangent is 1" Domain of Tanx is sin x tan x 1 cos x 3 7 x , 4 4 x 4 𝜋 − 2 <𝑥< 𝜋 2 Q4 *If the angle is in Q4, write it as a negative angle. Evaluate in radians without a calculator. 1. 𝐶𝑜𝑠 −1 − 3 2 cos𝑥 = − 𝐶𝑜𝑠 2. 𝑆𝑖𝑛 −1 −1 3 2 −1 and 𝑄2 3 5𝜋 − = 2 6 sin𝑥 = −1 and 𝑄4 𝑆𝑖𝑛−1 3. 𝑇𝑎𝑛−1 3 𝜋 −1 = − 2 tan𝑥 = 3 and 𝑄1 𝑇𝑎𝑛−1 𝜋 3 = 3 *If the angle is in Q4, write it as a negative angle. Hint: pay attention to restricted domain. −𝟏 𝒄𝒐𝒔 𝑻𝒂𝒏 𝟐 − 𝟑 Find the value of the above expression without a calculator. If the 𝑇𝑎𝑛𝜃 = 2 − 3 what is the cosθ? Tan domain is 𝜋 − 2 <𝜃< 𝜋 , 2 which is Quad I and Quad IV Since the tangent is negative, use Quadrant IV 1 2 Find cos Tan without a calculator. 3 "Cosine of the number 2 whose tangent is " 3 2 tan x 3 3 ADJ 3 13 2cos 94 c 13 HYP 13 c 13 3 13 2 Example 4 cos( Tan 1 2 B) Find the value of ) 3 with a calculator. No matter the mode setting is… Find the value without a calculator 1. 𝒄𝒔𝒄 𝟒 𝟑 −𝟏 𝟕 𝑪𝒐𝒔 𝟒 2. 𝒔𝒆𝒄 𝑺𝒊𝒏 −𝟏 − 𝟑 𝟏𝟑 𝟏𝟑 𝟐 Chapter 7 Test Thursday, March 2 No Calculator: Find exact values of trig functions Solve simple trig equations Given one trig function, find the others Know key points on graphs Find inverses of Sin, Cos, Tan Calculator: Sector area and arc length Reference angles Coterminal angles Convert degrees/radians Apparent size Find trig values from (x,y) point Homework Page 289 #1-13 odd Chapter Test Page 293 #1-12 No calculator on #6 - 12