MPM2D
Ms. Kueh
Right Angle Triangles
From a given angle, we can name the sides of a right angle triangle:
Adjacent is the side that touches the angle. Opposite is the side that does not touch the angle.
*Note: The opposite and adjacent sides depend on the angle! If we look at the same triangle
but a different angle, the opposite and adjacent sides will be different, but the Hypotenuse
always stays the same!
Identify the adjacent, opposite and hypotenuse sides for the angles in these right triangles:
Trigonometric Ratios
The following triangles are similar.
What do we know about the relationship between the sides of these triangles?
In fact, any right angle triangle with these same angles will have the _______________
_________________! We give these ratios special names:
Sine
Cosine
Tangent
These 3 are known as the ______________________ trig ratios.
Cosecant
Secant
Cotangent
These 3 are known as the ________________________________________________ trig ratios.
*These ratios Do Not Exist without the specified angle!
Also, sin 53.13° and sin 22.62° refer to different sets of similar triangles.
To remember the names of our primary trig ratios:
Example 1 For the triangle below, find
a) length of hypotenuse side
b) length of adjacent side
c) length of opposite side
d) sin x
e) cos x
f) tan x
Example 2 Calculate to 4 decimal places using your calculator.
*A note on Calculators* Make sure that your calculator is in Degrees, not Radians, or all your
numbers will be wrong!
a) sin 55
d) sin 115
b) cos 34
e) cos 90
c) tan 15
f) tan 96
Using Trigonometric Ratios to Find Sides
Example 3
1) First label the sides opposite, adjacent, hypotenuse.
2) Then look at the side we have, and the side we want to find.
3) Check
Hypotenuse.
Example 4 Find the side x.
Example 5
for which ratio has Adjacent and
Using Trigonometric Ratios to find Angles
Example 6 Calculate the angle x to the nearest degree
Now, we want x alone. To do this, we usually perform the opposite operation on x.
For example when we have 2x = 6,
2 is multiplying x, so we divide both sides by 2 to find x.
The opposite operation for sin is sin-1.
a) sin x = 0.707
d) sin x = 0.848
b) cos x = 0.259
e) cos x = 0.985
c) tan x = 1.732
f) tan x = -5.671
Example 6 Finding angle a.
Example 7
Summary
1) Always label the triangles first: opposite, adjacent, and hypotenuse
2) Figure out which ratio you need using
3) Solve for the unknown angle or side
Homework:
Worksheet pg. 496 #2,3,4,5
Textbook pg. 372 # 1EOO, 2EOO, 8, 9, 10abcd, 11abcd, 12