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. If OP is rotated counter-clockwise from the x-axis, the angle so formed is positive. But if OP is rotated clockwise from the x-axis, the angle so formed is negative. O P x negative angle P O x positive angle

(II)Degrees & Radians Angles are measured in degrees or radians. r r r 1c1c Given a circle with radius r, the angle subtended by an arc of length r measures 1 radian. Care with calculator! Make sure your calculator is set to radians when you are making radian calculations.

(III)Definition of trigonometric ratios x y P(x, y)  r y x Note: Do not write cos  1  tan  1 

From the above definitions, the signs of sin , cos  & tan  in different quadrants can be obtained. These are represented in the following diagram: All +ve sin +ve tan +ve 1st 2nd 3rd 4th cos +ve

What are special angles? (IV)Trigonometrical ratios of special angles Trigonometrical ratios of these angles are worth exploring 30 o, 45 o, 60 o, 90 o, …

1 sin 0°  0 sin 360°  0 sin 180°  0 sin 90°  1sin 270°  1 0   11

cos 0°  1 cos 360°  1 cos 180°  1 cos 90°  0 cos 270°  1 0   11

tan 180°  0 tan 0°  0 tan 90° is undefined tan 270° is undefined tan 360°  0 0  

Using the equilateral triangle (of side length 2 units) shown on the right, the following exact values can be found.

Complete the table. What do you observe?

2nd quadrant Important properties: 3rd quadrant 1st quadrant or 2     

Important properties: 4th quadrant or    or 2      In the diagram,  is acute. However, these relationships are true for all sizes of 

Complementary angles E.g.:30° & 60° are complementary angles. Two angles that sum up to 90° or radians are called complementary angles. are complementary angles. Recall:

Principal Angle & Principal Range Example: sin θ = 0.5 Principal range Restricting y= sinθ inside the principal range makes it a one-one function, i.e. so that a unique θ= sin -1 y exists

Example: sin. Solve for θ if Basic angle, α = Since sin is positive, it is in the 1 st or 2 nd quadrant Therefore Hence,

r y x A O P(x, y) By Pythagoras’ Theorem, (VI)3 Important Identities sin 2 A  cos 2 A  1 Since and, Note: sin 2 A  (sin A) 2 cos 2 A  (cos A) 2

(1) sin 2 A + cos 2 A  1(2) tan 2 A +1  sec 2 A(3) 1 + cot 2 A  csc 2 A tan 2 x = (tan x) 2 (VI)3 Important Identities Dividing (1) throughout by cos 2 A, Dividing (1) throughout by sin 2 A,

(VII)Important Formulae (1)Compound Angle Formulae

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,

(2)Double Angle Formulae (i) sin 2A = 2 sin A cos A (ii) cos 2A = cos 2 A – sin 2 A = 2 cos 2 A – 1 = 1 – 2 sin 2 A (iii) Proof:

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 mode to radians and use when you need to use degrees. 2nd o 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: 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: 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 Horizontal stretch or shrink; reflection about y -axis Horizontal shift Vertical shift Positive c moves left. Positive d moves up. The horizontal changes happen in the opposite direction to what you might expect. is a stretch. is a shrink.

When we apply these rules to sine and cosine, we use some different terms. Horizontal shift Vertical shift is the amplitude. is the period. A B C D

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.

Time: Pressure: nd {.00108,.00198,.00289,.00379, } STO alpha L 1 ENTER 2nd MATH 63 StatisticsRegressions 9 SinReg alpha L 1 L 2 ENTER Done The calculator should return:, Tuning Fork Data

2nd MATH 68 StatisticsShowStat ENTER The calculator gives you an equation and constants: 2nd MATH 63 StatisticsRegressions 9 SinReg alpha L 1 alpha L 2 ENTER Done The calculator should return:,

We can use the calculator to plot the new curve along with the original points: Y= y1=regeq(x) 2nd VAR-LINK regeq x ) Plot 1 ENTER WINDOW

Plot 1 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. These restricted trig functions have inverses. Inverse trig functions and their restricted domains and ranges are defined on page 47. 