Trigonometry Summary
Trigonometry gives us tools to solve ANGLES and SIDES in triangles. There are different tools for different
types of triangles.
RIGHT TRIANGLES
If we are trying to solve for sides or angles in right triangles, we will use the PRIMARY TRIGONOMETRIC
RATIOS.
The three primary trigonometric ratios are:
To be able to use these ratios, you need to be able to identify the HYPOTENUSE, OPPOSITE, and ADJACENT in
the triangle. Let’s try an example:
Example:
Label the side opposite the 20° angle, the side adjacent to the 20° angle, and the hypotenuse.
SOLVING FOR A SIDE/ANGLE USING THE PRIMARY TRIGONOMETRIC RATIOS
To be able to solve for a side, you must have at least
one angle and one side. Let’s try an example.
Example: Solve for the side labeled x.
To be able to solve for
an angle, you must
have at least TWO sides.
Let’s try an example
where we solve for an
angle.
Let’s solve for angle x.
Grade 10 Trigonometry Summary
ACUTE TRIANGLES
If the triangle is an acute triangle (all angles less than 90°), we cannot use the primary trigonometric ratios.
We need new tools. There are two LAWS that can be used to solve for sides or angles in acute triangles.
The SINE Law
The COSINE Law
Can only be used if we are given a side and angle
that are opposite each other and either another side
or angle.
Can only be used if:
-​ we are given two sides and the CONTAINED
angle (the angle between the two sides) or
-​ all three sides in the triangle.
What is the sine law?
The sine law works because the ratio of each side to
the SINE of its opposite angle is equal in any acute
triangle.
What is the cosine law?
Note: The side that starts the cosine law (in this
example side a) is OPPOSITE the angle in the cosine
law (in this example angle A).
Grade 10 Trigonometry Summary
ACUTE TRIANGLES - EXAMPLES
The SINE Law
The COSINE Law
Solving for a SIDE
Solve for side b in the triangle below.
Solve for side b in the triangle below.
Try a couple more for yourself - Solve for the indicated side.
1.​
2.​
3.​ In triangle KLM, ∠K=74°, ∠L=47.5°, and m =
37.7 cm. Find k.
4.​ In triangle CDE, ∠E=50°, c = 11.9 cm, and d =
13.5 cm. Find e.
ANSWERS: 1. 9.6 cm; 2. 8.6 cm; 3. 42.5 cm; 4. 10.8 cm
Grade 10 Trigonometry Summary
ACUTE TRIANGLES - EXAMPLES
The SINE Law
The COSINE Law
Solving for an ANGLE
Solve for angle A in the triangle below.
Solve for angle A in the triangle below.
Try a couple for yourself - Solve for the indicated angle.
1.​
2.​
ANSWERS: 1. 35°; 2. 55°
Grade 10 Trigonometry Summary