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In elementary geometry, an angle is a figure formed by ____ ______ with a common ______.
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The common endpoint of an angle is called the ______
vertex
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When three letter are used in this fashion <ABC, what can be said of B? *pretend < is the symbol for angle
It is the vertex
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Place the vertex of the angle at the center of a circle of radius r. Let s denote the length of the arc intercepted by the angle (figure 4 pg 424). The radian measure θ of the angle is the ratio of?
arc length s to the radius r
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What is the formula for radian measure θ
θ = s/r
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θ is measured in _______.
radians
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If θ (in a circle) is a right angle, then the arc length s is _____ ______ of the entire circumference. Derive the equation depicting this
- one quarter
- s = 1/4(2πr)
- s = πr/2
- *substitute value for s into s/r
- θ = (πr/2)/r
- θ = π/2radians = 90°
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Familiarize with the key positions on the unit circle
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Arc length Formula
s = rθ
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Sector Area Formula
A = 1/2r2θ
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If a radial line turns through an angle θ in time t, then the angular speed of the wheel ω (omega) is defined to be:
angular speed = ω = θ/t
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If a point P on the rotating wheel travels a distance d in time t, then the linear speed of P, denoted by v, is defined to be:
linear speed = v = d/t
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Convert 100 revolutions per minute to radians and explain
In each revolution there is 2π, so in 100rpm there'd be 100(2π) = 200π radians/min
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How do you find the linear speed of a point on the gear (rotating at 100rpm) 4cm from the center?
- You will use the linear speed formula = v = d/t
- To get d (s=rθ), you know that at 100rpm you have 200π radians, so you need to multiply that by 4cm
- This gives you d = 800πcm
- Now simply divide by 1min to get 800πcm/min
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Imagine an angle formed by two rays and accurately place the terminal side, initial side and the vertex
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We take the measure of an angle to be positive if the rotation is _______ and negative if the rotation is _______.
- counterclockwise
- clockwise
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In a rectangular coordinate system an angle is in _____ ______ if the vertex is located at (0,0) and the initial side of the angle lies along the _______ horizontal axis
- standard position
- positive horizontal axis
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P(x,y) denotes the point where the ______ side of angle θ intersects the unit circle
terminal side
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cosθ =
sinθ =
tanθ =
secθ =
cscθ =
cotθ =
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What is the formula for the unit circle
x2 + y2 = 1
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Where is sin cos tan positive vs negative on the unit circle
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θ = π/4 has a reference angle of ____ and coincides with the line _____. State the equation this sets up and its cos, sin and tan.
- 45°
- y = x
- x2 + x2 = 1 (if y = x)
- 2x2 = 1
- x =1/√2 = √2/2
- cos = x = √2/2
- sin = y = √2/2
- tan = y/x = 1
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θ = π/6. Derive your x and y values then state the cos, sin and tan. *Keep in mind these values are derived by forming an equilateral triangle by reflecting the line forming the angle across the x-axis
- y-(-y) = 2y
- we know the distance 2y = 1 (equilateral)
- therefore: y = 1/2
- x2 + y2 = 1
- x2 + 1/22 = 1
- x2 = 3/4
- x = √3/2
- cos = x = √3/2
- sin = y = 3/4
- tan = y/x = √3/3
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Which value for θ has the opposite values of (x,y) as does π/6. State the (x,y) values (with an explanation) and cos, sin and tangent
π/3 (1/2, √3/2), this is because the lines forming both reference angles are symmetrical about the line y = x
- cos = x = 1/2
- sin = y = √3/2
- tan y/x = √3
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Let θ be an angle in standard position, and suppose that θ is not a multiple of 90° or π/2. The reference angle associated with θ is the ______ angle (with _______ measure) formed by the ______ and the _______ side of the angle θ.
- acute angle
- positive measure
- x-axis & the terminal side
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When radian measure is used, the reference angle is sometimes referred to as the _______ number because a radian angle measure is a ______ number
- reference number
- real number
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In trying to find the reference angle for θ = 135°, we must first place the angle θ = 135° in _______ position on our unit circle. Then we find the _____ angle between the x-axis and the terminal side of θ. State the angle derived.
- standard position
- acute angle
- reference angle = 45°
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How do you conventionally write (sinθ) * (cosθ)? Likewise, how do you write 2(sin θ)? Notice the prevailing theme?
- sin θ cos θ
- 2 sin θ
- Omitting of the parenthesis
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Write (sin θ)2 conventionally
sin2 θ
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How would you show that cos A + cos B = cos(A + B) is not true in general
- (x + y)2 = x2 + y2 is usually untrue
- plug in values
- cos 30 + cos 60 = cos(30 + 60)
- √3/2 + 1/2 ≠ 0
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Trig Identity #1
sin2 θ + cos2 θ = 1
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sin θ/cos θ = tan θ is true for all real numbers except ______
zero
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Trig identity #2
sinθ/cosθ = tanθ
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Trig identity #3
- sin(90 - θ) = cos θ or sin(π/2 - θ) = cos θ
- cos(90 - θ) = sin θ or cos(π/2 - θ) = sin θ
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Give 2 examples of Trig identity #3
- sin 70° = cos 20°
- cos (3π/10) = sin (π/5)
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Using triangle <abc (with θ between b & c and β between a and c) prove sin2θ + cos2θ = 1
- sin2θ + cos2θ = 1 = (a/c)2 +(b/c)2
- a2/c2 + b2/c2
- a2 + b2/c2
- c2/c2
- 1
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Proof that sin(90° - θ) = cosθ
- Since the sum of the angles in any triangle is 180°:
- θ + β + 90° = 180°
- β = 90° - θ
- Since: sin(90° - θ) = sin β = b/c
- *note cos θ is also = b/c
- ∴ sin(90° - θ) = cos θ
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What makes two angles complementary, what does this mean for trig functions
- Their sum has to be 90°
- Significance: if two angles are complementary, then the sine of one equals the cosine of the other
- *incidentally, we get insight into the abbreviation "cosine" meaning complement's sine
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