
In elementary geometry, an angle is a figure formed by ____ ______ with a common ______.

The common endpoint of an angle is called the ______
vertex

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

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

What is the formula for radian measure θ
θ = s/r

θ is measured in _______.
radians

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°

Familiarize with the key positions on the unit circle

Arc length Formula
s = rθ

Sector Area Formula
A = 1/2r^{2}θ

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

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

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

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

Imagine an angle formed by two rays and accurately place the terminal side, initial side and the vertex

We take the measure of an angle to be positive if the rotation is _______ and negative if the rotation is _______.
 counterclockwise
 clockwise

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

P(x,y) denotes the point where the ______ side of angle θ intersects the unit circle
terminal side

cosθ =
sinθ =
tanθ =
secθ =
cscθ =
cotθ =

What is the formula for the unit circle
x^{2} + y^{2} = 1

Where is sin cos tan positive vs negative on the unit circle

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
 x^{2} + x^{2} = 1 (if y = x)
 2x^{2} = 1
 x =1/√2 = √2/2
 cos = x = √2/2
 sin = y = √2/2
 tan = y/x = 1

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 xaxis
 y(y) = 2y
 we know the distance 2y = 1 (equilateral)
 therefore: y = 1/2
 x^{2} + y^{2} = 1
 x^{2} + 1/2^{2} = 1
 x^{2} = 3/4
 x = √3/2
 cos = x = √3/2
 sin = y = 3/4
 tan = y/x = √3/3

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

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
 xaxis & the terminal side

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

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 xaxis and the terminal side of θ. State the angle derived.
 standard position
 acute angle
 reference angle = 45°

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

Write (sin θ)^{2} conventionally
sin^{2} θ

How would you show that cos A + cos B = cos(A + B) is not true in general
 (x + y)^{2} = x^{2} + y^{2} is usually untrue
 plug in values
 cos 30 + cos 60 = cos(30 + 60)
 √3/2 + 1/2 ≠ 0

Trig Identity #1
sin^{2} θ + cos^{2} θ = 1

sin θ/cos θ = tan θ is true for all real numbers except ______
zero

Trig identity #2
sinθ/cosθ = tanθ

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

Give 2 examples of Trig identity #3
 sin 70° = cos 20°
 cos (3π/10) = sin (π/5)

Using triangle <abc (with θ between b & c and β between a and c) prove sin^{2}θ + cos^{2}θ = 1
 sin2θ + cos2θ = 1 = (a/c)2 +(b/c)2
 a^{2}/c^{2} + b^{2}/c^{2}
 a^{2} + b^{2}/c^{2}
 c^{2}/c^{2}
 1

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

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

