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1.6 Trig Functions 1.6 Trig Functions The Mean Streak, Cedar Point Amusement Park, Sandusky, OH Trigonometry Review (I) Introduction By convention, angles are measured from the initial line or the x-axis with respect to the origin. P If OP is rotated counter-clockwise positive angle from the x-axis, the angle so formed x O is positive. But if OP is rotated clockwise from the x-axis, the angle so formed is negative. O x negative angle P (II) Degrees & Radians Angles are measured in degrees or radians. Given a circle with radius r, the angle subtended by an arc of length r measures 1 radian. r 1 c r rad 180 Care with calculator! Make sure your calculator is set to radians when you are making radian calculations. r (III) Definition of trigonometric ratios y P(x, y) r y sin sin cos tan hyp adj hyp opp adj 1 sin x x opp 1 Note: y cosec r x r y x sec sin cos 1 sin 1 Do not write 1 1 cos , tan . cos 1 cos cot tan sin From the above definitions, the signs of sin , cos & tan in different quadrants can be obtained. These are represented in the following diagram: sin +ve 2nd 3rd tan +ve All +ve 1st 4th cos +ve (IV) Trigonometrical ratios of special angles What are special angles? 30o, 45o, 60o, 90o, … , , , ,... 4 3 2 Trigonometrical ratios of these angles are worth exploring y sin x 1 0 2 1 sin 0 0 3 2 sin 2 0 sin 0 sin 0° 0 sin 1 2 sin 180° 0 sin 90° 1 2 sin 360° 0 3 sin 1 2 sin 270° 1 1 y cos x 0 2 1 cos 0° 1 cos 2 1 cos 360° 1 cos 1 cos 0 1 2 3 2 cos 180° 1 cos 0 2 cos 90° 0 3 cos 0 2 cos 270° 0 y tan x 0 tan 0 0 tan 0° 0 2 3 2 tan 0 tan 180° 0 2 tan 2 0 tan 360° 0 tan is undefined. 2 3 tan is undefined. 2 tan 90° is undefined tan 270° is undefined Using the equilateral triangle (of side length 2 units) shown on the right, the following exact values can be found. 1 sin 30 sin 6 2 3 sin 60 sin 3 2 1 cos 60 cos 3 2 3 cos 30 cos 6 2 1 tan 30 tan 6 3 tan 60 tan 3 3 1 2 sin 45 sin 4 2 2 cos 45 cos 4 1 2 2 2 tan 45 tan 1 4 Complete the table. What do you observe? Important properties: 2nd quadrant sin( ) sin 1st quadrant sin(2 ) sin cos( ) cos cos( 2 ) cos tan( ) tan tan(2 ) tan 3rd quadrant sin( ) sin cos( ) cos tan( ) tan Important properties: 4th quadrant sin(2 ) sin cos( 2 ) cos tan(2 ) tan sin() sin cos( ) cos tan() tan In the diagram, is acute. However, these relationships are true for all sizes of . Complementary angles Two angles that sum up to 90° or radians are called 2 complementary angles. E.g.: 30° & 60° are complementary angles. and are complementary angles. 2 Recall: 1 sin 30 cos 60 2 1 tan 30 cot 60 3 3 sin cos 3 6 2 tan 60 cot 30 3 Principal Angle & Principal Range Example: sinθ = 0.5 2 2 Principal range Restricting y= sinθ inside the principal range makes it a one-one function, i.e. so that a unique θ= sin-1y exists Example: sin Since ( 3 1 ) 2 2 3 sin ( ) 2 is positive, it is in the 1st or 2nd quadrant Basic angle, α = 4 3 4 Therefore 2 5 (inadmissib le ) 4 Hence, 3 4 . Solve for θ if 0 or 3 2 4 or 3 4 (VI) 3 Important Identities P(x, y) By Pythagoras’ Theorem, x2 y 2 r 2 2 2 x y 1 r r x y Since sin A and cos A , r r sin A2 cos A2 1 sin2 A cos2 A 1 O r y A x Note: sin 2 A (sin A)2 cos 2 A (cos A)2 (VI) 3 Important Identities (1) sin2 A + cos2 A 1 Dividing (1) throughout by cos2 A, tan 2 x = (tan x)2 (2) tan2 A +1 sec2 A 1 Dividing (1) throughout by sin2 A, (3) 1+ cot2 A csc2 A cos 2 A 1 cos A (sec A) 2 sec A 2 2 (VII) Important Formulae (1) Compound Angle Formulae 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 tan A tan B tan( A B) 1 tan A tan B tan A tan B tan( A B) 1 tan A tan B E.g. 4: It is given that tan A = 3. Find, without using calculator, (i) the exact value of tan , given that tan ( + A) = 5; (ii) the exact value of tan , given that sin ( + A) = 2 cos ( – A) Solution: (i) Given tan ( + A) 5 and tan A 3, tan tan A tan( A) 1 tan tan A tan 3 5 1 3 tan 5 15 tan tan 3 1 tan 8 (2) Double Angle Formulae (i) sin 2A = 2 sin A cos A Proof: sin 2 A (ii) cos 2A = cos2 A – sin2 A sin( A A) sin A cos A cos A sin A = 2 cos2 A – 1 = 1 – 2 sin2 A (iii) tan 2 A 2 tan A 2 1 tan A 2 sin Acos A cos 2 A cos( A A) cos 2 A sin 2 A 2 2 cos A (1 cos A) 2 cos 2 A 1 Trigonometric functions are used extensively in calculus. When you use trig functions in calculus, you must use radian measure for the angles. The best plan is to set the calculator o when you need to use mode to radians and use 2nd degrees. If you want to brush up on trig functions, they are graphed on page 41. Even and Odd Trig Functions: “Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change. Cosine is an even function because: cos cos Secant is also an even function, because it is the reciprocal of cosine. Even functions are symmetric about the y - axis. Even and Odd Trig Functions: “Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the sign of the y value also changes. Sine is an odd function because: sin sin Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function. Odd functions have origin symmetry. The rules for shifting, stretching, shrinking, and reflecting the graph of a function apply to trigonometric functions. Vertical stretch or shrink; reflection about x-axis a 1 is a stretch. Vertical shift Positive d moves up. y a f b x c d Horizontal shift Horizontal stretch or shrink; Positive c moves left. reflection about y-axis b 1 is a shrink. The horizontal changes happen in the opposite direction to what you might expect. When we apply these rules to sine and cosine, we use some different terms. A is the amplitude. Vertical shift 2 f x A sin x C D B Horizontal shift B is the period. B 4 A 3 C 2 D 1 -1 0 -1 2 y 1.5sin x 1 2 4 1 2 x 3 4 5 The sine equation is built into the TI-89 as a sinusoidal regression equation. For practice, we will find the sinusoidal equation for the tuning fork data on page 45. To save time, we will use only five points instead of all the data. Tuning Fork Data Time: Pressure: .00108 .200 .00198 .00289 .771 -.309 .00379 .480 .00108,.00198,.00289,.00379,.00471 L1 2nd ENTER { .00108,.00198,.00289,.00379,.00471 STO .2,.771, .309,.48,.581 L2 alpha .00471 .581 } 2nd L 1 ENTER ENTER SinReg L1, L2 ENTER 2nd MATH 6 Statistics 3 9 alpha SinReg Regressions L 1 , alpha The calculator should return: L 2 Done ENTER ExpReg L1, L2 ENTER 2nd MATH 6 Statistics 3 9 alpha SinReg Regressions L 1 , alpha The calculator should return: L 2 ENTER Done ShowStat ENTER 2nd MATH 6 Statistics 8 ENTER ShowStat The calculator gives you an equation and y a sin b x c d constants: a .608 b 2480 c 2.779 d .268 We can use the calculator to plot the new curve along with the original points: Y= 2nd Plot 1 y1=regeq(x) VAR-LINK x ) regeq ENTER ENTER WINDOW Plot 1 ENTER ENTER WINDOW GRAPH WINDOW GRAPH You could use the “trace” function to investigate the pressure at any given time. Trig functions are not one-to-one. However, the domain can be restricted for trig functions to make them one-to-one. 2 y sin x 3 2 2 2 3 2 2 These restricted trig functions have inverses. Inverse trig functions and their restricted domains and ranges are defined on page 47.