Worksheet 8 Properties of Trigonometric Functions Section Review of Trigonometry This section reviews some of the material covered in Worksheets 8, and The reader should be familiar with the trig ratios, using radians and working with exact values which arise from the following standard triangles π Example : Find the exact value of tan π y π π x π π 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 since sin θ is negative it must lie in the third or fourth quadrant looking at the standard triangle where sin θ = in the third and fourth quadrant we see that or θ = π + π = 5π θ = π π = 7π y x
Example : Find θ if cos θ = and π θ π y π/ π/ 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 quadrant note that π θ π so θ = π or θ = π Exercises: Find the exact values of the following trig ratios (a tan( 5π (b sin( 0π (c cos( 9π 5π (e sec( (d sin (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 π (e csc θ = where π θ π (f tan θ = where 0 θ π
Section Graphs of Trigonometric Functions Recall the graphs of the trig functions described below for π x π y = sin x 05 -Pi -Pi Pi Pi -05 - y = cos x 05 -Pi -Pi Pi Pi -05 - y = tan x -Pi -Pi Pi Pi - - From these graphs we can see some properties of these trig functions
These trig functions are periodic- they repeat themselves after a certain period sin x and cos x have period π: ie sin x = sin(x + π x R cos x = cos(x + π x R tan(x has period π: ie 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, k Z 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 x R tan( x = tan x x R Since cos x is an even function we have cos( x = cos x x R These graphs alter if we change the 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
Example : Sketch y = sin x for 0 x π Here the period is now π = π y = sin x 05 Pi Pi -05 - Example : Sketch y = sin x for 0 x π Here the period is still π but the amplitude is now y = sin x - Pi Pi - - Example : Sketch y = sin ( x + π for π x π The period is π, the amplitude is and there is a phase shift The graph shifts to the left by π y = sin ( x + π - Pi Pi - - 5
Exercises: Sketch the following graphs stating the period for each (a y = cos x (b y = cos ( x π (c y = tan ( x + π (d y = + sin ( x π (e y = cos ( x + π (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 cos x 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 (x, y θ x 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 = cos θ + sin θ = ie
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 sin(a + B = sin A cos B + cos A sin B sin(a B = sin A cos B cos A sin B cos(a + B = cos A cos B sin A sin B cos(a B = cos A cos B + sin A sin B Proof of - : Consider the following circle of radius with angles A and B as shown Q(cos A, sin A d P (cos B, sin B A B 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 d = + (( cos(a B = cos(a B Equating these we see that cos(a B = cos A cos B + sin A sin B, 7
thus proving sum and difference identity We can use identity 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 (A + B A B = cos = cos A = sin A cos B + cos cos B + sin A = sin A cos B + cos A sin B sin B A sin 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 - Double Angle Identities cos θ = cos θ sin θ = sin θ = cos θ : sin θ = sin θ cos θ Proof of and : 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 (Pythagoeran identity we can see that cos θ = sin θ Replacing sin θ by cos θ (Pythagoeran identity we can see that cos θ = cos θ 8
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 and Half Angle Identities : cos ( θ : sin ( θ = + cos θ = cos θ To prove the half angle identities we begin by rearranging the double angle identities Proof of : We take the double angle identity cos θ = cos θ to obtain cos θ = cos θ + cos θ + ie cos θ = ie cos ( θ = cos θ + Proof of : We take the double angle identity cos θ = sin θ to obtain sin θ = cos θ cos θ ie sin θ = ie sin ( θ We have now established identities 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 = sin x cos x = sin (x (double angle = sin x 9
Example : Show that cos x cos x sin x = cos x cos x cos x sin x = cos x (cos x + sin x = cos x (Pythagorean identity = cos (x (double angle = cos x 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 x 5 Using Pythagoras we see that the horizontal side is We must have cos x = 5, since cosine is negative in the second quadrant So sin x = sin x cos x = ( ( 5 5 = 5 Example : Use the appropriate half angle identity to find the exact value of sin ( 5π 8 The half angle identity for sine is ( θ sin = cos θ ( θ cos θ ie sin = ± 0
Now, sin 8 ( π/ = sin cos (π/ = ± / = ± Since π 8 = ± = ± = ± lies in the first quadrant, where sine is positive, we must have sin = 8 Exercises: Simplify the following (a cos 5x sin x cos x sin 5x (b sin x + cos x + tan x sec x (c sin (x/8 (d cot x sin x cos x cos x sin x (e sin x cos x sin x cos x (f ( cos (x/ + sin (x/ Use the addition formulas to find the exact value of the following ( ( ( 7π π 7π (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 8 π Given tan x = 5 for 0 x π, evaluate the following (a sin x (b cos x (c sin x (d cos x (e cos x
Exercises 8 Properties of Trigonometric Functions Solve the following equations (a tan x =, π x π (c sin x sin x = 0, 0 x π (b sin x =, 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 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 cos x cos x cos x + sin x sin x sin x (d = cos x sin x sin x cos x Suppose α lies in the third quadrant, β lies in the fourth quadrant and sin α = 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 Use the half angle identity to find the exact value of sin( π 8 7 Use the appropriate identities to find the exact values of the following ( (a cos π ( (b sin + π
Answers 8 Section (a (b (c (d (e (f (a 5π, 7π (b π, 7π (c π, π (d π, 5π, π, π (e 5π (f π, π, 5π, 7π Section (a period is π y = cos x 05 -Pi -Pi Pi Pi -05 - (b period is π y = cos ( x π -Pi -Pi Pi Pi - -
(c period is π ( y = tan x + π -Pi -Pi Pi Pi - - (d period is π ( x π y = + sin 5 05 -Pi -Pi Pi Pi (e period is π y = cos ( x + π 5 5 05 -Pi -Pi Pi Pi
(f period is π y = cos x 08 0 0 0 -Pi -Pi Pi Pi (g period is π y = cos x 05 -Pi -Pi Pi Pi -05 - (a π, π, 095, 5 (c 0, π, π, 5π, 7π, π (b π, π, 5π (d π, 5π Section (a sin x (b ( x (c cos (d (e sin x (f (a (b (c (a + (b + (c 5
(a 5 (b (c 0 9 (d 9 9 (e 7 75 Exercises 8 (a π, π π (b, 5π ( x (a y = cos (c 0, π, π, π, π, π, 5π (d 0, π, π, 5π -Pi -Pi Pi Pi - - ( (b y = + sin x + π 5 05 -Pi -Pi Pi Pi
( (c y = tan x π -Pi Pi - - Proofs only (a 7 5 (b 75 (c 7 5 (a 9 5 (b 5 7 7 (a + + + 8 (b 7