4.11.4 Trigonometry

4.11.4 Trigonometry

• 1.
  4.11.4 Trigonometry The studentis able to (I can): For any right triangle • Define the sine, cosine, and tangent ratios and their• Define the sine, cosine, and tangent ratios and their inverses • Find the measure of a side given a side and an angle • Find the measure of an angle given two sides • Use trig ratios to solve problems
• 2.
  By the Angle-AngleSimilarity Theorem, a right triangle with a given acute angle is similar to every other right triangle with the same acute angle measure. This means that the ratios between the sides of those triangles are always the same. Because these ratios are so useful, they were given names: sinesinesinesine, cosinecosinecosinecosine, andwere given names: sinesinesinesine, cosinecosinecosinecosine, and tangenttangenttangenttangent. These ratios are used in the study of trigonometry.
• 3.
  sine sine of∠A AAAA hypotenuse adjacent opposite leg opposite A sinA hypotenuse ∠ = = cosine tangent cosine of ∠A tangent of ∠A hypotenuse leg adjacent to A cosA hypotenuse ∠ = = leg opposite A tanA leg adjacent to A ∠ = = ∠
• 4.
  We can usethe trig ratios to find either missing sides or missing angles of right triangles. To do this, we will set up equations and solve for the missing part. In order to figure out the sine, cosine, and tangent ratios, we can use either a calculator or a trig table.
• 5.
  To use theNspire calculator to find tan 51°: • From a New Document, press the µ key: • Use the right arrow key ( ) to select tan• Use the right arrow key (¢) to select tan and press ·:
• 6.
  • Type 5Iand hit ·: To use the calculator on your phone: • Turn your phone landscape to access the scientific calculator. • Type the angle in first, thenthenthenthen select tan.
• 7.
  To use atrig table to find cos 52°: • Locate 52° on the table. • Scan over to the Cos column and find the value. • cos 52° = .6157
• 8.
  To find anangle, we use the inverseinverseinverseinverse trig functions (in more advanced classes, you will hear them referred to as arcsine, arccosine, and arctangent). On your calculator, these are listed as sin—1, cos—1, and tan—1. Ex. Find :−       1 8 sin 17 Press the µ button, and then the ¤ arrow to select sin—1. Then enter 8p17·. You should get 28.07… This means that the angle opposite a leg of 8 with a hypotenuse of 17 will measure around 28˚.    17
• 9.
  To find anangle using a trig table, just find the appropriate trig column, find the closest value, and read back to the angle. Ex. Find ( )−1 tan 0.35 0.35 is closer to 0.3443 than it is to 0.3640, so our answer would be 19˚.
• 10.
  You will beexpected to memorize these ratio relationships. There are many hints out there to help you keep them straight. The most common is SOHSOHSOHSOH----CAHCAHCAHCAH----TOATOATOATOA , where A mnemonic I like is “Some Old Hippie pO S p in pHy = dA C j os pHy = pO T p an jAd = A mnemonic I like is “Some Old Hippie Caught Another Hippie Trippin’ On Acid.” Or “Silly Old Hitler Couldn’t Advance His Troops Over Africa.”
• 11.
  Examples I. Usethe triangle to find the following ratios. 1. sin A = _____ A B C 8 15 17 1. sin A = _____ 2. cos A = _____ 3. tan A = _____
• 12.
  Examples I. Usethe triangle to find the following ratios. 1. sin A = _____ A B C 8 15 17 15 171. sin A = _____ 2. cos A = _____ 3. tan A = _____ 8 17 17 15 8
• 13.
  Examples I. Usethe triangle to find the following ratios. 4. sin B = _____ A B C 8 15 17 8 174. sin B = _____ 5. cos B = _____ 6. tan B = _____ 17 15 17 8 15
• 14.
  Examples II. Findthe lengths of the sides to the nearest tenth. 1. x (opp) 15 (adj) 58° ° = = ° ≈ x tan58 15 x 15tan58 24.0 2. 26 (hyp) x (adj) 46° ° = = ° ≈ x cos46 26 x 26cos46 18.1
• 15.
  III. Find themissing angle to the nearest whole degree. 26 (hyp) 19 (opp) xº ° = 19 sinx 26 −   =     1 19 x sin 26 ≈ °x 47