Triangle Definition of sin θ and cos θ

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1 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.

2 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.

3 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.

5 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.

6 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. MATH 8 Lecture A 6 of 5 Ronald Brent 7 All rights reserved.

7 You should commit to memor the following conversion chart since these angles will come up again and again. Degrees 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.

8 (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.

9 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.

10 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.

12 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.

13 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.

14 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.

15 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.

16 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.

17 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 = MATH 8 Lecture A 7 of 5 Ronald Brent 7 All rights reserved.

23 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.

C.3 Review of Trigonometric Functions

C.3 Review of Trigonometric Functions C. Review of Trigonometric Functions C7 C. Review of Trigonometric Functions Describe angles and use degree measure. Use radian measure. Understand the definitions of the si trigonometric functions. Evaluate