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An angle can be measured in revolutions, degrees and radians.

Angle Ratio

Angle Ratio

What to Learn:

Help to Remember What to Learn

 

 

 

Diagram

initial arm: for an angle in standard position, the arm along the positive x-axis

terminal arm: for an angle in standard position, the arm that is free to rotate

standard position: the location of an angle in the plane in which the vertex is at the origin, the initial arm lies along the positive x-axis, and the terminal arm is free to rotate

positive angle: an angle in standard position swept out by a counterclockwise rotation of its terminal arm

negative angle: an angle in standard position swept out by a clockwise rotation of its terminal arm

positive angle(°) - negative angle(°) = 360°

Interactive Activity

  • Locate the sliders below.
  • Move the slider(s) to generate positive and negative angles.
  • Notice that the sliders move together. This happens because:

    positive angle(°) - negative angle(°) = 360°

  • Check the vocabulary/concepts in the "What to Learn" column.

Diagram

principal angle: the smallest positive angle

coterminal angles: angles in standard position with the same initial arm and terminal arm as the principal angle. Coterminal angles are found by adding or subtracting the principal angle by a multiple of 360°.

coterminal(°) = principal(°) + n(360°)

quadrant of terminal arm:

  • Quadrant 1(Q1): 0 < P < 90°
  • Quadrant 2(Q2): 90° < P < 180°
  • Quadrant 3(Q3): 180° < P < 270°
  • Quadrant 4(Q4): 270° < P < 360°

Interactive Activity

  • Move the slider(s) to generate principal and coterminal angles.

    coterminal angle(°) = principal angle(°) + 360n(°)

  • Check the vocabulary/concepts in the "What to Learn" column.

reference angle: the acute angle formed by the terminal arm of an angle in standard position and the x-axis

  • Q1: reference(°) = principal(°)
  • Q2: reference(°) = 180° - principal(°)
  • Q3: reference(°) = principal(°) - 180°
  • Q4: reference(°) = 360° - principal(°)

Interactive Activity

  • Move the slider to generate principal and reference angles.
  • Q1: reference(°) = principal(°)
  • Q2: reference(°) = 180° - principal(°)
  • Q3: reference(°) = principal(°) - 180°
  • Q4: reference(°) = 360° - principal(°)
  • Check the vocabulary/concepts in the "What to Learn" column.

primary trigonometry ratios: sine, cosine, tangent

Right Triangle Definitions

      

CAST is a memory aide to help remember the primary trigonometric ratios that are positive in each quadrant.

  • All primary trigonometry ratios are positive in Q1
  • Sine is only positive ratio in Q2
  • Tangent is only positive ratio in Q3
  • Cosine is only positive ratio in Q4
 Primary Ratio
Q1
Q2
Q3
Q4
sin= y/r
+/+ = +
+/+ = +
-/+ = -
-/+ = -
cos= x/r
+/+ = +
-/+ = -
-/+ = -
+/+ = +
tan= y/x
+/+ = +
-/+ = -
-/- = +
-/+ = -
 memory aide
A
S
T
C

Interactive Activity

  • Move the slider(s) to genenerate x- and y-values.
  • Study the r calulation in the middle of the screen
  • Study the primary trigonometry ratios on the right
  • Study which ratios are positive in each of the 4 quadrants (Q1, Q2, Q3, Q4)

reciprocal function: a function of the form obtained by taking the reciprocal of the function y = f(x)

reciprocal trigonometry ratios: secant, cosecant, cotangent

sin A = reciprocal

cos A = reciprocal

tan A = reciprocal

Reciprocal Ratio
Q1
Q2
Q3
Q4
csc= reciprocal
+/+ = +
+/+ = +
+/- = -
+/- = -
sec= reciprocal
+/+ = +
+/- = -
+/- = -
+/+ = +
cot= reciprocal
+/+ = +
+/- = -
-/- = +
+/- = -

Interactive Activity

  • Move the slider(s) to genenerate x- and y-values.
  • Study the r calulation in the middle of the screen
  • Compare the pairs of primary and reciprocal trigonometry ratios on the right
  • Study which ratios are positive in each of the 4 quadrants (Q1, Q2, Q3, Q4)

 

Review: trigonometry ratios, angles and sides in quadrant 1. This activity uses opposite, adjacent and hypotenuse. When we look at all 4 quadrants it is more convenient to use y, x and .

sin A = image, find other five trigonometry ratios

State known sides, calculate third side: x = -2, r = 3, y = image= image

csc A = image = image

cos A =image

sec A = image

tan A = image

cot A = image

Interactive Activity

  • "Introduction" reviews the "other rules" for finding sides/angles in a right triangle.
  • "Naming Sides" provides practice for naming the sides of a right triangle. Pairs of these sides form the trigonometric ratios.
  • "Practice" - The first problem types deal with the Pythagorean theorem. The other 15 types deal with primay trigonometric ratios, sides and angles.

Additional Resource:

http://www.learnalberta.ca/content/mejhm/html/object_interactives/trigonometry/explore_it.html

Math 30P   Angles   Arcs   Unit Circle   Sine/Cosine Transformations  Sine/Cosecant Transformations   Trigonometry Review

Comments to:  Jim Reed - Homepage