Text has been moved down to match the height of the interactive element on the right.

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.

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°

• 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(°)

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

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

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

reciprocal trigonometry ratios: secant, cosecant, cotangent

sinA = 1/cscA

cosA = 1/secA

tanA = 1/cotA

Reciprocal Ratio	Q1	Q2	Q3	Q4
csc= r/y	+/+ = +	+/+ = +	+/- = -	+/- = -
sec= r/x	+/+ = +	+/- = -	+/- = -	+/+ = +
tan= x/y	+/+ = +	+/- = -	-/- = +	+/- = -

• 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 r.

• "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