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8.2 Trigonometry
- 1. Trigonometry The student is able to (I can): For any right triangle • 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-Angle Similarity 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, and tangenttangenttangenttangent. These ratios are used in the study of trigonometrytrigonometrytrigonometrytrigonometry.
- 3. sine of ∠A cosine of ∠A tangent of ∠A AAAA hypotenuse adjacent opposite ∠ = = leg opposite sin hypotenuse A A leg adjacent to cos hypotenuse A A ∠ = = leg opposite tan leg adjacent to A A A ∠ = = ∠
- 4. Examples I. Use the triangle to find the following ratios. 1. sin B = _____ 2. cos B = _____ 3. tan B = _____ A B C 8 15 17
- 5. Examples I. Use the triangle to find the following ratios. 1. sin B = _____ 2. cos B = _____ 3. tan B = _____ A B C 8 15 17 8 17 15 17 8 15
- 6. To use the Nspire calculator to find tan 51°: • From a New Document, press the µ key: • Use the right arrow key (¢) to select tan and press ·:
- 7. • Type 5I and hit ·: To use the calculator on your phone: • Turn your phone landscape to access the scientific calculator. • Depending on your phone, you will either teither teither teither type the angle in first and select tan, orororor select tan and then type in the angle.
- 8. To find an angle, we use the inverseinverseinverseinverse trig functions (you will sometimes 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 : 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˚. 1 8 sin 17 −
- 9. You will be expected 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 Caught Another Hippie Trippin’ On Acid.” Or “Silly Old Hitler Couldn’t Advance His Troops Over Africa.” pp in yp O S H = dj os yp A C H = pp an dj O T A =
- 10. We can use the trig ratios to find either missing sides or missing angles of right triangles. To do this, we set up equations and solve for the missing part. When setting up your equation, use the following pattern: where “trig” is sine, cosine, or tangent ratio, A is the angle, and the two sides are the ones that go with that ratio. Now solve for the missing piece. ( )° = side trig side A
- 11. II. Find the lengths of the sides to the nearest tenth. 1. 2. x (opp) 15 (adj) 58° 26 (hyp) x (adj) 46° ( ) ( ) ° = = ° ≈ tan 58 15 15tan 58 24.0 x x ( ) ( ) ° = = ° ≈ cos 46 26 26cos 46 18.1 x x (x on top: multiply)
- 12. 3. Also, notice that the sides are multiplying/dividing the trig ratio, notnotnotnot the angle. x (hyp) 16 (opp) 37° ( ) ( ) ( ) ° = ° = = ° ≈ 16 sin 37 sin 37 16 16 sin 37 26.6 x x x (x on bottom: divide)
- 13. III. Find the missing angle to the nearest whole degree. 26 (hyp) 19 (opp) x° 1 19 sin 26 19 sin 26 47 x x − ° = = ≈ ° You’ll learn more about this in Algebra II, but a function and its inverse cancel each other out. So ( )− ° = °1 sin sinx x