Properties of Trigonometric functions
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1 Worksheet 4.6 Properties of Trigonometric functions Section Review of Trigonometry This section reviews some of the material covered in Worksheets.,. and.4. The reader should be familiar with the trig ratios, using radians and working with exact values which arise from the following standard triangles. π 6 π 4 π π 4 Example : Find the exact value of tan π. y π x π lies in the third quadrant and the angle made with the horizontal axis is π. tan is positive in the rd quadrant looking at the corresponding standard triangle in the third quadrant we see that tan( π = + Example : Find θ if sin θ = and 0 θ π. y x since sin θ is negative it must lie in the third quadrant o fourth quadrant looking at the standard triangle where sin θ = in the third and fourth quadrant we see that θ = π + π 4 = 5π 4 or θ = π π 4 = 7π 4
2 Example : Find θ if cos θ = and π θ π. y π/6 π/6 x since cos θ is positive it must lie in the first or fourth quadrant look at the standard triangles where cos θ = in the first and fourth quadrants note that π θ π so θ = π 6 or θ = π 6 Exercises:. Find the exact values of the following trig ratios. (a tan( 5π 9π (c cos( 4 5π (e sec( 6 (b sin( 0π (d sin( 4π 6 (f cot( π. Find the value of θ in the following exercises. (a cos θ = where 0 θ π (b tan θ = where 0 θ π (c sin θ = where π θ π (d sec θ = where 0 θ 4π (e csc θ = where π θ π (f tan θ = where 0 θ π 4
3 Section Graphs of trigonometric functions Recall the graphs of the trig functions described below for π x π.
4 We can see some properties of the trig functions from their graphs. These trig functions are periodic they repeat themselves after a certain period. sin x and cos x both have period π. i.e. sin x = sin(x + π cos x = cos(x + π x R x R tan x has period π. i.e. tan x = tan(x + π x R Note that sin x and cos x both lie between and. Note that tan x is undefined for x = (k π when k Z. 4 From the graphs we can see that ( sin x + π ( π cos x = cos x = sin x 5 Since sin x and tan x are odd functions we have sin( x = sin x tan( x = tan x x R x R Since cos x is an even function we have cos( x = cos x x R These graphs alter if we change either the period or amplitude, or if there is a phase shift. Consider the graph of y = sin x. In general we can think of this as y = A sin n(x a, where A is the amplitude n alters the period (period = π n by subtracting a from x, the graph shifts to the right by a. So y = sin x has amplitude and period π. 4
5 Example : Sketch y = sin x for 0 x π. Here the period is now π = π. Example : Sketch y = sin x for 0 x π. Here the period is still π but the amplitude is now. Example : Sketch y = sin (x + π 4 for π 4 x π. The period is π, the amplitude is and there is a phase shift. The graph shifts to the left by π 4. 5
6 Exercises:. Sketch the following graphs and state the period for each. (a y = cos 4x (b y = cos 4(x π (c y = tan (x + π (d y = + sin( x π (e y = cos(x + π 6 (f y = cos x (g y = cos x. Solve the following equations for 0 x π. (a cos x cos x = 0 (b sin x + sin x = 0 (c 4 cos x 4 cos x cos x + = 0 (d tan x + tan x + = 0 Section Trigonometric identities This section states and proves some common trig identities. Pythagorean Identities cos θ + sin θ = + tan θ = sec θ + cot θ = csc θ Proof of : Consider a circle of radius centred at the origin. 6
7 x (x, y y Let θ be the angle measured anticlockwise for the positive x-axis. Using trig ratios we see that x = cos θ and y = sin θ. By Pythagoras Theorem x + y =. i.e. cos θ + sin θ =. Proof of : Divide both sides of identity by cos θ and the result follows. Proof of : Divide both sides of identity by sin θ and the result follows. Sum and Difference Identities 4 sin(a + B = sin A cos B + cos A sin B 5 sin(a B = sin A cos B cos A sin B 6 cos(a + B = cos A cos B sin A sin B 7 cos(a B = cos A cos B + sin A sin B Proof of 4 7 : We first prove 7. Consider the following circle of radius with angles A and B as shown. y Q(cos A, sin A d P (cos B, sin B A B x 7
8 Note that we can label point P as (cos B, sin B and point Q as (cos A, sin A by using trig ratios. We can calculate the distance d using two methods. Using the distance formula we see that d = (cos B cos A + (sin B sin A = cos B cos A cos B + cos A + sin B sin A sin B + sin A = (cos B + sin B + (cos A + sin A (cos A cos B + sin A sin B = (cos A cos B + sin A sin B Using the cosine rule we have Equating these we see that d = + (( cos(a B = cos(a B thus proving sum and difference identity 7. cos(a B = cos A cos B + sin A sin B We can use identity 7 to deduce the remaining identities. We have cos(a + B = cos(a ( B = cos A cos( B + sin A sin( B = cos A cos B sin A sin B sin(a + B = cos ( π (A + B = cos ( π A B = cos ( π A cos B + sin ( π A sin B = sin A cos B + cos ( π ( π A sin B = sin A cos B + cos A cos B sin(a B = sin(a + ( B = sin A cos( B + cos A sin( B = sin A cos B cos A sin B We have now established identities
9 Double Angle Identities 8 cos θ = cos θ sin θ = sin θ = cos θ 9 sin θ = sin θ cos θ Proof of 8 and 9 : using the sum and difference identities we can prove the double angle identities. For instance, cos θ = cos(θ + θ cos θ cos θ sin θ sin θ = cos θ sin θ Replacing cos θ by sin θ (Pythagorean identity, we can see that cos θ = sin θ. Replacing sin θ by cos θ (Pythagorean identity, we can see that cos θ = cos θ. We also have sin θ = sin(θ + θ = sin θ cos θ + cos θ sin θ = sin θ cos θ. We have now established identities 8 and 9. Half Angle Identities ( θ 0 cos ( θ sin = + cos θ = cos θ To prove the half angle identities we begin by rearranging the double angle identities. Proof of 0 : We take the double angle identity cos θ = cos θ to obtain cos θ = cos θ + cos θ + i.e. cos θ = ( θ i.e. cos 9 = cos θ +
10 Proof of : We take the double angle identity cos θ = sin θ to obtain i.e. i.e. sin ( θ sin θ = cos θ cos θ sin θ = We have now established identities 0 and. = cos θ Example : Simplify cos x cos x sin x sin x sin x sin x. cos x cos x sin x sin x sin x sin x = sin x(cos x cos x sin x sin x (factorise = sin x(cos(x + x (difference identity 6 = sin x cos x = sin (x (double angle identity 9 = sin 6x Example : Show that cos x cos x sin x = cos 4x. cos x cos x sin x = cos x (cos x + sin x = cos x (Pythagorean identity = cos (x (double angle identity 8 = cos 4x Example : Given sin x = for π x π, find: (i cos x and (ii sin x. 5 Since π x π, we know x lies in the second quadrant. Also, since sin x =, we 5 can form the following right angled triangle. 5 x Using Pythagoras we see that the horizontal side is 4. We must have cos x = 4, 5 since cosine is negative in the second quadrant. So sin x = sin x cos x = ( 5 ( 4 5 = 4 5 0
11 Example 4 : Use the appropriate half angle identity to find the exact value of sin( π 8. The half angle identity for sine is is Now, i.e. ( θ sin = cos θ ( θ cos θ sin = ± ( π sin 8 ( π/4 = sin cos(π/4 = ± / = ± Since π 8 = ± = ± 4 = ± lies in the first quadrant, where sine is positive, we must have ( π sin = 8.
12 Exercises:. Simplify the following (a cos 5x sin x cos x sin 5x (b sin x + cos x + tan x sec x (c 6 sin (x/8 cot x sin x cos x (d cos x sin x (e sin x cos x 4 sin x cos x ( (f cos(x/ + sin (x/4. Use the addition formulas to find the exact value of the following. ( 7π ( 4π ( (a cos (b sin (c tan. Use the half angle identities to calculate the exact value of the following. ( ( ( (a cos (b cos (c sin π 8 4. Given tan x = 5 for 0 x π, evaluate the following. π (a sin x (b cos x (c sin x (d cos x (e cos x 7π 6 π
13 Exercises for Worksheet 4.6. Solve the following equations. (a tan x =, π x π (b sin x =, 0 x π (c 4 sin x sin x = 0, 0 x π (d sec x sec x + = 0, 0 x π. Sketch the following curves. ( x (a y = cos ( (b y = + sin x + π ( (c y = tan x π 6. Prove the following. (a cos x sin x cos x sin x = sin x (b tan x sin x tan x = sin x cos x (c sin x = + sin x cos x 4. Suppose α lies in the third quadrant, β lies in the fourth quadrant and sin α = 4 with 5 cos β =. Find the following. 5 (a sin(α + β (b cos(α + β (c tan(α + β 5. Suppose cos x = where π (a cos x x π. Find the exact value of the following. (b sin x 6. Use the half angle identity to find the exact value of sin( π Use the appropriate identities to find the exact values of the following. ( ( π (a cos 4 π ( ( π (b sin 6 + π 4
14 Answers for Worksheet 4.6 Section. (a (c (e (b (d (f. (a 5π 6, 7π 6 (b π 6, 7π 6 (c π 6, π (d π 4, 5π 4, π 4, π 4 (e 5π 6 (f π 4, π 4, 5π 4, 7π 4 Section. (a period is π (b period is π (c period is π 4
15 (d period is 6π (e period is π (f period is π (g period is π 5
16 . (a π, π, 0.95, 5.4 (b π, π 6, 5π 6 (c 0, π, π 6, 5π 6, 7π 6, π 6 (d π 4, 7π 4 Section. (a sin 4x ( x (c cos 4 (e sin 4x (b (d (f 0. (a (b + (c. (a 4. (a 5 (b (b (c 0 69 (d (c (e Exercises 4.6. (a π, π π (b, 5π (c 0, π, π, π, π, 4π, 5π (d 0, π, π, 5π. (a y = cos( x 6
17 (b y = + sin(x + π (c y = tan ( (x π 6. Proofs only. 4. (a (a 9 54 (b 4 75 (b (c (a (b 7
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