Trigonometry Table of Contents

Math 20P - Trigonometry

Pure Math 20 Program of Studies (2002, Alberta, Canada)

Students are expected to solve problems involving triangles, including those found in 3-D and 2-D applications. (M)
6.1 Solve problems involving two right triangles. [CN, PS, V]
(M) 6.2 Extend the concepts of sine and cosine for angles from 0° to 180°. [R, T, V]
(M) 6.3 Apply the sine and cosine laws, excluding the ambiguous case, to solve problems.[CN, PS, V]

The tools provided on this page can be applied to solving problems involving two right triangles.

Labeling triangles: For triangle ABC, the angles are <A, <B, <C and the sides a, b, c are opposite their corresponding angles.

Remember that the sum of angles that form a triangle or line is 180^o. If the triangle is a right triangle, one angle is always 90^o. The remaining two angles must total (180 - 90)^o = 90^o.

If <A = 25 and <B = 75. Then the third angle in triangle ABC is [180 - (25 + 75)]^o = 80^o.
If acute angle A in a right triangle ABC is 45^o, the other acute angle is (90 - 45)^o = 45^o.
Consider line segment AC. If <AB and <BC form this line, the sum of <AB and <BC is 180^o. If we know <AB is 50^o, then <BC must be 130^o.

If you know 2 sides of a right triangle, the third side may be determined using the Pythagorean Theorem.

You will need to understand the angle of depression equals the angle of depression.

The resource below shows two parallel lines and a transversal (intersecting line). To model the angle of elevation equals the angle of depression using congruent alternate angles, select Alternate and Reference Angle C. Move the Animation Slider to see that <F is congruent to <C. The half turn animation may help you to remember why the angle of depression equals the angle of depression.

Note: The angle of elevation equals the angle of depression is commonly used in trigonometry problems. If you are given the angle of elevation, then show the angle of depression on your diagram as well.

Link: LearnAlberta

What to Learn:

Help to Remember What to Learn

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.

Note: A and B are the acute angles in a right triangle.The adjacent and opposite sides when A is the reference angle become the opposite and adjacent sides when B is the reference angle.

Therefore:

sinA = opposite/hypotenuse = cos(90 - A) = cosB

cosA = adjacent/hypotenuse = sin(90 - A) = sinB

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 primary trigonometric ratios, sides and angles.

Additional Resources:

The Law of Sines: apply when you know the requirements for:

1 angle & the side opposite the angle plus a second angle or side.

The Law of Cosines: apply when you know the measurements for:

3 sides

2 sides and the angle between these known sides.
a² = b² + c² - 2bc cosA

b² = a² + c² - 2ac cosB

c² = a² + b² - 2ab cosC

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.

Pure Math 10: Extend the concepts of sine and cosine for angles from 0° to 180°. The following resource extends this to 0° to 360°.

primary trigonometry ratios: sine,cosine, tangent

Right Triangle Definitions

For Pure Math 10 focus on Q1 and Q2.

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

Note: For Pure Math 10 focus on Q1 and Q2. Students in this course will not be expected to use CAST in their solutions.

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 generate x- and y-values.
Study the r calculation 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), but for Pure Math 10 focus on Q1 and Q2.

Comments to: Jim Reed - Homepage