100 Unit 7: Trigonometry Part 1 Right Triangle Trigonometry Hypotenuse a) Sine sin( α ) = d) Cosecant csc( α ) = α Adjacent Opposite b) Cosine cos( α ) = e) Secant sec( α ) = c) Tangent f) Cotangent tan( α ) = cot( α ) = Ex. 1: Find the values of the six trigonometric functions of the angleθ. 7 θ Ex.: Find the exact values of the sin,cos, and tan of 45 45
101 Ex. : Find the exact values of the sin,cos, and tan of 60 and 0 0 Ex. 4: Find the exact value of x (without a calculator). 5 0 x Ex. 5: Find all missing sides and angles (with a calculator). 1
10 Applications An angle of elevation and an angle of depression can be measured from a point of reference and a horizontal line. Draw two figures to illustrate. Ex. 6: (use a calculator) A surveyor is standing 50 feet from the base of a large building. The surveyor measures the angle of elevation to the top of the building to be 71.5. How tall is the building? Draw a picture. Ex. 7: A ladder leaning against a house forms a 67 angle with the ground and needs to reach a window 17 feet above the ground. How long must the ladder be?
10 Angles Degrees and Radians. An angle consists of two rays that originate at a common point called the vertex. One of the rays is called the initial side of the angle and the other ray is called the terminal side. Angles that share the same initial side and terminal side are said to be coterminal. To find a co-terminal angle to some angle α (in degrees): Radians the other angle measure. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Radians are a dimensionless angle measure.
104 Conversion between degrees and radians 60 = radians. Therefore, a) 1 radian = degrees b) 1 = radians Ex. 1: Convert the following radian measure to degrees: a) 5 6 b) 10 c) 4 d) Ex. : Convert the following degree measure to radians: a) 400 b) -10 To find a co-terminal angle to some angle α (in radians): Ex. : Find a coterminal angle, one positive and one negative, to 5.
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106 We use the unit circle to quickly evaluate the trigonometric functions of the common angle found on it. To summarize how to evaluate the Sine and Cosine of the angles found on the unit circle: 1. sin(α ) =. cos(α )= Ex. 1: Find the exact value a) sin( ) 4 b) cos( ) c) sin( ) 6 e) sin(150 ) d) cos( ) f) 11 cos( ) 6 g) sin( ) h) 5 i) sin(0 ) cos( ) Ex. : Find the exact value by finding coterminal angles that are on the unit circle. a) 1 sin( ) 4 b) 7 sin( ) c) sin( 00 ) 6
107 Cosine is an even function. Sine is an odd function. cos( α) = cos( α) sin( α) = sin( α) Ex. : Find the exact value. a) 7 cos( ) b) 6 sin( ) c) 4 1 cos( ) 4 Reference Angles For any angle α in standard position, the reference angle ( α ' ) associated with α is the acute angle formed by the terminal side of α and the x - axis. Ex. 1: Find the reference angle α ' for the given angles. a) α = b) α =. c) 5 α = d) 4 α = 5
108 The signs (+ or value) of the Sine, Cosine and Tangent functions in the four quadrants of the Euclidean plane can be summarized in this way: Ex.: Find the exact value. a) sin( ) b) 5 cos( ) 6 c) 5 sin( ) d) cos( ) e) cos( 00 ) f) sin(150 ) Ex.: Find all values of θ in the interval [0, ] that satisfy the given equation a) sin( θ ) = b) cos( θ ) = Ex. 5: If sin( t ) = and t, find the value of cos( t ).
109 Arc Length In a circle of radius r, the length s of an arc with angle θ radians is: s = rθ Ex. 1: Find the length of an arc of a circle with radius 5 and an angle 5. 4 Ex. : Find the length of an arc of a circle with radius 1 and an angle0. Ex. : The arc of a circle of radius associated with angle θ has length 5. What is the measure of θ? Area of a Circular Sector In a circle of radius r, the area A of a circular sector formed by an angle of θ radians is A= 1 r θ Ex. 1: Find the area A of a sector with angle 45 in a circle of radius 4.
110 Graphs of the Sine and Cosine Functions Ex. 1: Graph f( x) = sin x Domain: Range: x -intercepts: Period: Amplitude: Even or odd? Ex. : Graph f( x) = cos x Domain: Range: x -intercepts: Period: Amplitude: Even or odd?
111 Ex.: Graph one period of each function a) f( x) = cos( x) b) f( x) = 1+ sin( x) c) f( x) = cos( x ) d) f( x) = sin( x)
11 Graphs of f ( x) = Asin( Bx + C) + D and f ( x) = Acos( Bx + C) + D where A 0 and B > 0 Amplitude: Period: Horizontal shift (Phase shift): Vertical shift: have: Ex.1: Graph one period of each function. a) f( x) = sin( x ) Amplitude: Period: Horizontal Shift: End Points: b) 1 f( x) = sin( x+ ) 4 Amplitude: Period: Horizontal Shift: End Points:
11 c) f( x) = 1+ cos( x ) 4 Amplitude: Period: Horizontal Shift: End Points: Ex. : Write a Sine or Cosine function whose graph matches the given curve. a) x -scale is 4 b) x -scale is Ex. : Write a Sine and Cosine function whose graph matches the given curve. x -scale is