The Trigonometric Functions

  1. In elementary geometry, an angle is a figure formed by ____ ______ with a common ______.
    • two rays
    • endpoint
  2. The common endpoint of an angle is called the ______
    vertex
  3. 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
  4. 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
  5. What is the formula for radian measure θ
    θ = s/r
  6. θ is measured in _______.
    radians
  7. 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°
  8. Familiarize with the key positions on the unit circle
    Image Upload 1
  9. Arc length Formula
    s = rθ
  10. Sector Area Formula
    A = 1/2r2θ
  11. 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
  12. 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
  13. 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
  14. 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
  15. Imagine an angle formed by two rays and accurately place the terminal side, initial side and the vertex
    Image Upload 2
  16. We take the measure of an angle to be positive if the rotation is _______ and negative if the rotation is _______.
    • counterclockwise
    • clockwise
  17. 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
  18. P(x,y) denotes the point where the ______ side of angle θ intersects the unit circle
    terminal side
  19. cosθ =
    sinθ =
    tanθ =
    secθ =
    cscθ =
    cotθ =
    • x
    • y/x
    • 1/x 
    • 1/y
    • x/y
  20. What is the formula for the unit circle
    x2 + y2 = 1
  21. Where is sin cos tan positive vs negative on the unit circle
    Image Upload 3
  22. 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
  23. 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
  24. 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
  25. 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
  26. 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
  27. 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°
  28. 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
  29. Write (sin θ)2 conventionally
    sin2 θ
  30. 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
  31. Trig Identity #1
    sin2 θ + cos2 θ = 1
  32. sin θ/cos θ = tan θ is true for all real numbers except ______
    zero
  33. Trig identity #2
    sinθ/cosθ = tanθ
  34. 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
  35. Give 2 examples of Trig identity #3
    • sin 70° = cos 20°
    • cos (3π/10) = sin (π/5)
  36. 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
  37. 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
  38. 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
Author
chikeokjr
ID
333343
Card Set
The Trigonometric Functions
Description
Chapter 6
Updated