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Primary and Reciprocal Trig Ratios The primary trig ratios refer to the relationships between the sides and non-right angles in right-angled triangles. opp hyp adj cos hyp opp tan adj sin In standard position sin θ = cos θ = tan θ = The reciprocal trig ratios are as follows csc 1 sin sec 1 cos cot 1 tan sec hyp adj cot adj hyp and therefore it follows that : csc hyp opp To find trig ratios, your calculator must be in the correct mode (degrees or radians) Ex. 1 Find the following ratios rounded to 4 decimal places. a) cos(1.2164) b) tan( 0.4387) c) csc(1.6493) It is important to recognize that it is the angle that you are taking the sin/cos/tan/csc/sec/cot of, and the ratio of the sides in a related right angled triangle you are finding. There are several special angles we can memorize the ratios for (Special triangles in grade 11). We refer to these as the related acute angles (R.A.A or R ) as we move forward in this unit. We memorize the ratios of these acute angles. Special Angle #1. 4 These ratios can also be used to find exact trig ratios of angles sketched in other quadrants when Ex. 2 Find cos and csc of the angle Special Angle # 2. Again, the ratios of the angle 7 . 4 Special Angle # 3. 6 is the R.A.A. 4 3 and can be used anytime an angle has a R.A.A. of or respectively. 6 3 6 3 Ex. 3 Sketch the following angles in standard position, and use your knowledge of the R.A.A. to find all 6 trig ratios for each. 2 9 a. b. 3 4 Ex. 4 Find θ, given 0o ≤ θ ≤ 90o 1 a. sin 2 b. sec 2 3 More Angles in Standard Position Recall: We can evaluate trig functions for angles greater than 90o or radians by drawing the angle in 2 standard position and using the related acute angle (RAA). Using the CAST rule Ex. 1 Determine the exact measure of the indicated ratio for the given angle. Ex.2 Determine the exact measure of the following using example 1 as a reference. Ex. 3 The coordinates of a point on the terminal arm of a standard position angle, θ are (-2, 6). Find all six trig ratios for this angle. Sometimes, there is difficulty with finding trig ratios of angles that fall on one of the axes in standard position. If we draw our angles in standard position as normal and think about the x, y and r values, we can determine the corresponding trig ratios. Homework: 1. Evaluate. a) 2. sin120 b) tan 150 sec 120 c) d) 2 3 e) Evaluate each of the following. Give exact answers. a) sin g) sin 6 7 b) tan 6 k) 2 sin h) 2 cos 2 6 cos 6 5 4 cos c) 4 h) l) 6 1 i) cos 2 3 tan 3 csc 2 cos i) cot 3 4 2 sin cos m) sin 4 4 2 2 cos j) 1 2 sin 3 2 3 d) tan 3 sin 540 sin 5 6 j) 4 2 n) cos k) sin 3 e) f) cos 6 sin csc 390 cos sec 2 3 4 3 3 o) cos 2 3 3. The coordinates of a point on the terminal arm of a standard position angle, θ are (5, -12). Find all six trig ratios for this angle. 4. Determine sin and cot if csc 4 3 and 0 2 . Answers to Homework Graphs the Sine and Cosine Functions Complete the following table for the function y = sinx and graph the function on the grid provided. Characteristics of y = sinx Key Points Domain Maximum Value Minimum Value Range Period Amplitude Complete the following table for the function y = cosx and graph the function on the grid provided. Characteristics of y = cosx Key Points Domain Maximum Value Minimum Value Range Period Amplitude Transformations of sin and cosine functions y a sin k ( x p) q y a cos k ( x p ) q a k p q Ex. 3 Write two possible equations for the graph shown. y 5 x π 2π More Sine & Cosine Graphs and the Tangent Function Recall: From Last Day The base sine and cosine functions appear as follows To graph transformations of these functions consider the effects of the following variables Note: the angle must be in factored form to determine k Graphing the Tangent Function, y = tan Consider x, y and r values to determine the values of tan θ when θ is an angle on one of the axes. In order to graph y = tan θ, we must consider the behaviour of the function around the undefined values which represent vertical asymptotes on our graph. Using these characteristics we draw the tangent function Characteristics of the Tangent Function Graphs of Reciprocal Trigonometric Functions How to graph the reciprocal functions: 1) Determine where the reciprocal function will intersect the primary function. 2) Determine and plot where Vertical asymptotes will be located. This will occur when the function is undefined (denominator = 0). 3) Determine the behaviour of the graph as the function approaches vertical asymptotes. The graph of y = csc x Below is a graph of the function y = sin x. The reciprocal of this function is y = csc x, where csc x 1 sin x . 1) Functions intersect at ___________________________ 2) Vertical Asymptotes at ___________________________ 3) Behaviour around the asymptotes Properties of y = csc x Domain Range Period The graph of y = sec x Below is a graph of the function y = cos x. The reciprocal of this function is y = sec x, where sec x 1 cos x 1) Functions intersect at ___________________________ 2) Vertical Asymptotes at ___________________________ 3) Behaviour around the asymptotes Properties of y = sec x Domain Range Period The graph of y = cot x Below is a graph of the function y = tan x. The reciprocal of this function is y = cot x, where cot x 1 tan x 1) Functions intersect at 2) Vertical Asymptotes at __________________________ 3) Behaviour around the asymptotes *** Also consider zeros of y = cot x Properties of y = cot x Domain Range Period Work: Sketch two periods of each of the following the following. Scale in radians. 1. y tan x 1 4. y sec( x ) 4 y csc x 3 7. 2. 5. 8. y 2sec x y csc x 2 3. 1 y tan x y csc( x ) 1 6. 2 2 y 2 tan 3 x 1 (challenge) 4 Solving Trigonometric Equations Some Tools for solving trig equations: Ex. 1 Solve each equation for 0 2 . a) sin c) sec 2 1 2 2 0 b) 2cos 3 0 d) tan 2 1 Ex. 2 Solve for 0 x 360 b) 3 sec x – 5 = 0 a) cos x = 0.75 Practice Questions 1. 2. 3. Solve. 0 2 a) 3sin 1 sin e) tan 1 h) sin 1 0 j) cos 1 0 Solve. a) d) g) 0 x 360 3sin x 2 0 7cos x 5 1 3sin x 4 cos 1 0 f) sec 2 cot 3 i) k) csc 1 0 b) b) e) h) 2csc x 5 0 cot x 5 0 cos 2x 1 2cos 3 0 g) 3csc 6 0 c) l) tan c) f) I) 2tan x 7 2sec x 3 9 sec x 3 2 c) 2sin Solve 0 2 a) 2 sin 2 1 b) 3 cos 3 2 2 1 0 d) 2 cos 1 Solving Trig Equations Part II Recall: Solving trig equations requires knowledge of Special Triangles and the CAST rule. In some cases, we can also use our knowledge of the graphs of trig functions to help to solve equations. The graphs of y = sin x, and y = cos x are shown below for convenience. y 1 x π/2 π 3π/2 2π π/2 π 3π/2 2π -1 y 1 x -1 Ex. 1 Solve for where 0 2 a) sin 2 1 0 b) 2 cos 2 5cos 2 0 c) 4sin 2 7 sin 3 0 Math 12 pg. 320 # 4ac, 11, 12, 14ac, 15 ac, 16 Applications of Trigonometric Functions Day 1 Ex. 1 In a harbour, the water depth at high tide is 6.3m and low tide is 2.3m. High tide occurs at 12 noon while low tide occurs at 6:12 p.m. Determine (using a sketch of the graph if helpful): a) the mean depth b) period c) amplitude d) an equation using the sine function that represents the depth of water in the harbour at time t e) the depth of the water at 1 p.m. f) when a ship that requires a depth of 3.3m of water can safely enter the harbour Applications of Trigonometric Functions Day 2 Ex. 1 A Ferris wheel with a 10m diameter rotates once every 30 seconds. Passengers get on at the lowest point of the ride (1m off the ground). i. Draw a graph showing height above the ground through the first two cycles. ii. Write an equation (using a sine function) to express height as a function of time. iii. Calculate the height of the ride after 9s. iv. Between what times will the Ferris wheel be 9m or higher in the first revolution? pg 369 # 6 - 9, 11a,15