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AP CALCULUS AB 5.6 Inverse Trigonometric Functions Objectives: Develop properties of the six inverse trig functions and differentiate an inverse trig function Inverse Functions • To have an inverse, a function must be one to one and have a reflection across the line y = x. • However, none of the 6 trig functions have an inverse because they are periodic and not one to one. • In order for the trig functions to have an inverse, the domains and ranges must be restricted. • Take a look at the Inverse Trig worksheet for the Domain and Range of each function. Restrictions for an Inverse sin 𝑦 = 𝑥 𝑖𝑓𝑓 𝑦 = 𝑎𝑟𝑐 sin 𝑥 𝑦 = 𝑠𝑖𝑛 or −1 y = 𝑠𝑖𝑛−1 𝑥 1 𝑥≠ sin 𝑥 Example 1 𝑦 = 𝑎𝑟𝑐 sin − 2 1 sin 𝑦 = − 2 Sine of what angle is 1 equal to − ? 2 𝜋 𝑦=− 6 Example 𝑦 = 𝑎𝑟𝑐 cos 0 cos 𝑦 = 0 Cosine of what angle is equal to 0? 𝜋 𝑦= 2 Example 𝑦 = 𝑎𝑟𝑐 tan 3 𝑡𝑎𝑛 𝑦 = 3 sin 𝑦 3 = cos 𝑦 1 3 sin 𝑦 = 2 1 cos 𝑦 2 𝜋 𝑦= 3 Example with Calculator 𝑦 = 𝑎𝑟𝑐 sin(0.3) MODE must be in RADIANS 𝑦 = 𝑠𝑖𝑛−1 (0.3) 𝑦 ≈ 0.305 Example with Calculator 𝑦 = 𝑎𝑟𝑐 sin(𝜋) MODE must be in RADIANS 𝑦 = 𝑠𝑖𝑛−1 (𝜋) 𝑦 ≈ 𝐸𝑅𝑅𝑂𝑅 WHY? 𝜋 𝜋 − ≤𝜃≤ 2 2 SOH – CAH – TOA • Given: 𝑦 = 𝑎𝑟𝑐 sin 𝑥 • Find: cos 𝑦 𝑦 = 𝑎𝑟𝑐 sin 𝑥 𝑥 𝑂𝑝𝑝 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 sin 𝑦 = = 1 𝐻𝑦𝑝 • cos 𝑦 = cos(𝑎𝑟𝑐 sin 𝑥) • Use Right Triangle Trig to Solve • cos 𝑦 = • cos 𝑦 = 𝐴𝑑𝑗 𝐻𝑦𝑝 1−𝑥 2 1 Opp = x y Adj = b 𝑥 2 + 𝑏2 = 12 𝑏2 = 1 − 𝑥 2 𝑏 = ± 1 − 𝑥2 𝑏 = 1 − 𝑥2 sin 𝑦 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑠𝑜 + 1 − 𝑥 2 SOH – CAH – TOA • Given: 𝑦 = 𝑎𝑟𝑐 s𝑒𝑐 5 2 • Find: tan 𝑦 tan 𝑦 = tan 𝑎𝑟𝑐 sec 5 2 • Use Right Triangle Trig to Solve • tan 𝑦 = • tan 𝑦 = 𝑂𝑝𝑝 𝐴𝑑𝑗 1 2 5 𝑦 = 𝑎𝑟𝑐 s𝑒𝑐 2 5 𝐻𝑦𝑝 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 s𝑒𝑐 𝑦 = = 2 𝐴𝑑𝑗 Opp = b y Adj = 2 22 + 𝑏2 2 = 5 4 + 𝑏2 = 5 𝑏2 = 1 𝑏 = ±1 s𝑒𝑐 𝑦 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑠𝑜 𝑏 = +1 Formative Assessment • Pg. 377 (5-19) odd