Triangle Definition of sin θ and cos θ

Triangle Definition of sin θ and cos θ Then Consider the triangle ABC below. Let A be called θ. A HYP (hpotenuse) θ ADJ (side adjacent to the angle θ ) B C OPP (side opposite to the angle θ ) (SOH CAH TOA) OPP sin θ = = HYP BC AB ADJ AC cos θ = = HYP AB. MATH 8 Lecture A of 5 Ronald Brent 7 All rights reserved.

Special Angles (, 5, 6 ) Find sin 5 and cos 5. 5 s Pthagorean Theorem: s + s =, or s = Hence s = and s=±. 5 s Choosing s >, we have s =, and so sin 5 = cos 5 = = 5 5 MATH 8 Lecture A of 5 Ronald Brent 7 All rights reserved.

Eample: Find sin, cos, sin 6, and cos 6. Begin with a 9 6 right triangle with HYP =. s 6 t 6 Notice that when flipping the triangle down and consider the larger one, the result is an equilateral triangle, so the vertical side is also. This means t =, or s + t =, we have s + =, t =. Now since MATH 8 Lecture A of 5 Ronald Brent 7 All rights reserved.

Angles in Radian Measure How big is a radian? Here s how big: it s the angle corresponding to an arc length of in a unit circle. Look at the diagram below. A unit circle indicates that the radius = unit, and we ll alwas put the center at (,) for convenience. The angle θ as drawn is radian, because the arc length subtended (cut off) b the angle has length = unit. Length of the arc = unit θ = radian (,) (,) unit MATH 8 Lecture A 5 of 5 Ronald Brent 7 All rights reserved.

Relationship Between Degrees and Radians 6 = radians rad = 6 57 = rad = rad.7 rad 6 8. Eamples: a) Convert to radians. = 8 radians = 8 =. 6 Note: 5 =, 6 = and 9 =. 5 b) Convert radians to degrees. 6 = 8 5 6 5 6 = 8 = 5. MATH 8 Lecture A 6 of 5 Ronald Brent 7 All rights reserved.

You should commit to memor the following conversion chart since these angles will come up again and again. Degrees 6 7 8 9 6 5 Radians 6 Consider the unit circle, centered at the origin, with an angle of θ radians, as shown below. (cos θ, sin θ ) θ (,) MATH 8 Lecture A 7 of 5 Ronald Brent 7 All rights reserved.

(Notice that the angle is measured from the positive -ais, counterclockwise.) The dotted line defining the terminal side (end) of the angle θ intersects the circle at a point. As the angle θ changes, so do the coordinates of that point, so each of the coordinates is a function of the angle θ. These two functions are ver important, and so the have their own names. Definition: In the figure below, the first coordinate is called cos θ (short for cosine of θ ). The second coordinate is called sin θ ( short for sine of θ ). (cos θ, sin θ ) θ (,) MATH 8 Lecture A 8 of 5 Ronald Brent 7 All rights reserved.

Remarks: a) Since this point is on the unit circle, its coordinates must satisf the equation of that circle: + =, that is (cosθ ) + (sinθ ) =. b) To avoid the constant use of brackets, we write cos n θ to mean (cos θ ) n ; similarl, we write sin n θ to mean (sin θ ) n. Thus cos θ + sin θ =. c) Since the ccle repeats ever time we go around the circle, the sine and cosine functions are periodic with period. MATH 8 Lecture A 9 of 5 Ronald Brent 7 All rights reserved.

Table of Trig. Values: Since = radians, 6 5 = radians, and 6 results and the following picture to fill in the table on the net page. = radians, we can use the previous triangle trig. (,) ( cosθ,sinθ ) (-,) (,) (,-) MATH 8 Lecture A of 5 Ronald Brent 7 All rights reserved.

Trig Values at Other Angles: Definition: Eample: Find Since An Obtuse angle is an angle greater than radians. sin and cos. =, the triangle in the figure below is the triangle shown previousl. Hence, and cos sin = =. 6 9, 6 = (,) MATH 8 Lecture A of 5 Ronald Brent 7 All rights reserved.

Definition: A negative angle is an angle measured in the clockwise direction. Eample: Find sin and cos. Since = 6 shown is again the shown above. Hence, sin =, the triangle in the figure 6 9 triangle and cos =. 6 (,) MATH 8 Lecture A of 5 Ronald Brent 7 All rights reserved.

Graphs of the sine and cosine functions (cos θ, sin θ ) = sinθ θ (,) θ = cosθ The second coordinate, = sin θ, goes from to, and back down to, then back θ up to. Meanwhile, the first coordinate, goes from down to, and back to. = cosθ, MATH 8 Lecture A of 5 Ronald Brent 7 All rights reserved.

Graphs of Sine and Cosine (Sinusoidal) Functions = sin Notice how these graphs oscillate between and. Also, the length, or period, of one full ccle is. = cos MATH 8 Lecture A 5 of 5 Ronald Brent 7 All rights reserved.

Formal Definitions An function of the form = k sin( a φ ) + C or = k cos( a φ ) + C is called sinusoidal. The Amplitude of a sinusoidal graph is equal to one-half the distance from the top to the bottom of the waves, or the number k. The Period of a sinusoidal function is the distance for the graph to go through one full ccle. It is alwas P =. a The Angular (Circular) Frequenc of a sinusoidal function, a, is the number of complete ccles in a horizontal distance of. The Linear Frequenc, f is the reciprocal of the a period, so that f =. If represents time, then f has units of ccles per second. The Phase φ of a sinusoidal function is what point in its ccle it starts at, when =. It represents horizontal shifts in the sinusoidal function. The horizontal line = C, is called the center line about which the function oscillates. MATH 8 Lecture A 6 of 5 Ronald Brent 7 All rights reserved.

Eample: The graph below is ( ) = sin + f. Its amplitude is k = =, (NOT the bigger number 5.) The circular frequenc is, the period is, and the frequenc is. The phase is, and the center line is =. 5 - - MATH 8 Lecture A 7 of 5 Ronald Brent 7 All rights reserved.

Of course all this vertical amplitude scaling works for the cosine graph also. = cos 5 = cos 5 - - - - - - - - -5-5 = cos - - - - MATH 8-5 Lecture A -5 of 5 Ronald Brent 7 All rights reserved. 5 = cos 5 - - - -

Changes in Frequenc: Going from = sin or = cos, to = sin( a ) and = cos( a ) involves horizontal scaling. This affects how man ccles appear over a given interval. As a rule: (a) If a is a positive integer, then the graph of = sin( a ) ( = cos( a ) ) has a complete oscillations, or ccles, in the interval [, ]. For a positive, if a > this means more oscillations than = sin ( = cos ) and for < a <, one has less ccles than = sin ( = cos ). (b) If a <, the graph is reflected about the -ais, and then compressed or stretched depending upon the value of a. Note: = sin( a ) and = a sin are NOT the same. Test it with a = and =. MATH 8 Lecture A of 5 Ronald Brent 7 All rights reserved.

Phase Shifts: Phase shifts involve horizontal translations, of shifts in the -direction. Eamples: = sin ( ) = cos( + ) 5 5 - - - - - - - - -5-5 MATH 8 Lecture A 5 of 5 Ronald Brent 7 All rights reserved.