Sec 2 Hon Notes RIGHT TRIANGLE Trigonometry 8.1. adjacent _ leg cos hypotenuse

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1 Sec Hon Notes RIGHT TRIANGLE Trigonometry 8.1 Trig Ratios: sine, cosine, tangent, cosecant, secant, cotangent (theta): acute angle opposite _ leg sin hypotenuse adjacent _ leg cos hypotenuse opposite _ leg tan adjacent _ leg csc hypotenuse opposite _ leg sec hypotenuse adjacent _ leg adjacent _ leg cot opposite _ leg Memory Trick: SOH - CAH -TOA Trig ratios only apply to right triangles! Find the trig ratios: Find missing side lengths where necessary (reduced root form). Give all ratios in reduced fraction form where appropriate R M L s t T r S sin S = csc S = cos S = sec S = tan S = cot S = N O sin O = csc O = cos M = sec M = tan M = cot O = K P sin K = csc L = cos L = sec K = tan K = cot L = 4. Find the trig ratios. Remember: trig ratios only apply to right triangles! Hint: You will need to add a segment to the diagram Y 5. For these problems you will need to draw your own triangle. Hint: use the simplest possible side lengths appropriate for the triangle. Rationalize your ratios. (No roots left in denominator.) 5 5 X 14 Z sin 45 = csc 45 = cos X = cot Z = sec Z = csc Z = tan X = sin X = cos 30 = sec 30 = tan 60 = cot 60 =

2 Calculator Trigonometry ALWAYS CHECK the MODE Round to 3 decimal places!!! Finding the trig RATIOS: Why do we need a calculator for these ratios when we didn t for the last problem set? 1. sin37. tan53 3. sec8 (Write down the calculator steps for this.) Finding the ANGLE!! How do you do this on the calculator? 4. If sin A = , then A = 5. If cos A = 0.091, then A = 6. If sec A = 1.156, then A = (Write down the calculator steps for this!!!) Find the missing values. Show neat, complete work!!! 7. A boat travels in the following path. How far north did it travel? 8. An architect needs to use a diagonal support in an arch. Her company drew the following diagram. How long does the diagonal support have to be? 9. Rennie is walking her dog. The dog s leash is 1 feet long and is attached to the dog 10 feet horizontally from Rennie s hand, as shown in the diagram. What is the angle formed by the leash and the horizontal at the dog's collar? 10. A park has a skateboard ramp with a length of 14. feet and a length along the ground of 1.9 feet. The height is 5.9 feet. Calculate the measure of the angle formed by the ramp and the ground. When giving an angle measure in degrees, you must have a degree symbol!!!

3 Angles & Special Triangles Sec Hon Notes RIGHT TRIANGLE Trigonometry 8. (Orally review the trig ratio formulas as a class) Define the following terms: Standard Position Angle: Initial ray: Terminal ray: Coterminal angles: Reference angles: 1 st : Draw all the angles in standard position. nd: Find, and label the measure of, the reference angle º. 40º º º º º º 8. Give two angles coterminal with 10º 3

4 Reference Triangles For each angle measure below, sketch a coordinate system showing all possible angle measures (4) with the given reference angle measure. Then construct the reference triangle for each and label the side lengths. (Hint: The leg lengths may be negative, but the hypotenuse will always be a positive length) Look at Geogebra Unit Circle 30 degrees 60 degrees 45 degrees 11. Find the following trig ratios by drawing triangles (No Calculator!) on a coordinate system. Rationalize answers. a. sin 10 b. cos 10 c. tan 330 d. csc 135

5 Sec Hon Notes RIGHT TRIANGLE Trigonometry 8.3 Unit Circle!! sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = Find the value of the following: 1. cos 60 =. sin 10 = 3. tan 90 = 4. cot 90 = 5. sec 60 = 6. sec 60 = 7. csc 135 = 8. cos 60 = 9. tan 300 = 10. sec 0 = 11. tan 70 = 1. cot 180 = 13. sin 40 = 14. csc 540 = 15. sec 855 = Draw triangles to find the following. No Calculator! Give all answers in reduced root/fraction form. 1. cot300 csc300. cot 330 csc If θ is in standard position and contains the point (-8, 15), find sin θ and sec θ. 4. If θ is in standard position and contains the point (1, 14) find tan θ. 5

6 Sec Hon Notes RIGHT TRIANGLE Trigonometry 8.4 Radians (180º = π radians) Think of radians as slicing a circle into fractions. It is very important that you get comfortable working and thinking in radians!!! Complete the diagram below by adding the angle measures in π form of radians, and decimal radians. Example: 90 = π , Convert from degrees to radians. Write your answer in reduced fraction form in terms of π. Try to find the value of the angle by looking at the diagram before calculating it º. -180º 3. 30º º º 6. 70º Convert from radians to degrees. Remember degree marks! Try to find the value of the angle by looking at the diagram before calculating it. 7. π π Convert from degrees to radians in decimal form ( decimal places). Look at the diagram at the top of this page and try to estimate the value before calculating it º 1. 16º 6

7 Convert from radians to degrees in decimal form ( decimal places). Look at the diagram at the top of the notes and try to estimate the value before calculating it. How do you know these angles are in radians? Review: Find exact answers (trig ratios) for the following. NO Calculator. Include a labeled triangle with your ratio. 15. sin cos cot sec Review: Find the trig ratios in decimal form to two decimal places. Calculator review. Hint: given angles are in radians. How do you know the values are angles? How do you know the angles are in radians? 19. sin csc (- 1.34) 7

8 Sec Hon Notes RIGHT TRIANGLE Trigonometry 8.5 (QUIZ day) Inverse Trig Functions: Given the ratio, find the angle measure/s Use your unit circle to find θ 1. If sin θ = 1 find: a b c. no restrictions. If sin θ = 3 find: a. 0 b. 0 c. no restrictions 3. If cos θ = find: a b. 3 c. no restrictions Inverse (Arc) Trig functions. Given the ratio, find the angle/s. Arc s have specific domains or quadrants (called principal values) where you look to find the solutions. Trig function Principal Values: or 0 Arc sin or sin -1 : or Arc cos or cos -1 : Arc tan or tan -1 : or Draw a triangle to find the following. Write all answers in BOTH degrees and radians. 4. Arc sin (-1/) = 6. Arc tan 1 = 7. Sin -1 = 8

9 Sec Hon Notes RIGHT TRIANGLE Trigonometry 8.6 Identities: true for all angles. Pythagorean Identities Reciprocal Identities Negative Identities sin cos 1 1tan sec 1cot csc 1 csc sin 1 sec cos 1 cot tan sin( ) sin cos( ) cos tan( ) tan csc( ) csc sec( ) sec cot( ) cot Ratio Identities Cofunction Identities cos sin sin tan cos csc sec cos cot sin cot tan Prove the following identities: sin cos sec csc tan cot 1. sin sec csc tan. sec tan (1 tan ) 3. Simplify cos xsin x (sin x1)(sin x1) **REMEMBER** *Don t invent new rules. *Changing things to sin and cos usually works. *You can t use Pythagorean unless things are squared. *Don t move things across the equal sign 9 when proving identities.