Download Special Triangles Investigation

Math 20-1
Trigonometry
Name: _____________________
Special Triangles Investigation
Specific Outcome
T 1: Demonstrate an understanding of angles in
standard position (0° to 360°).
T 2: Solve problems, using the three primary
trigonometric ratios for angles from 0° to 360°
in standard position.
Questions
2, 5, 8
Mark
9
1, 3-4, 6-7,
9
18


In this investigation, you will discover the exact value of the sine, cosine, and tangent ratios of
some special angles.
Exact value means that you express your answer as an integer, fraction, or radical, not as a
rounded decimal. Here are a few examples:
3
6
15 5
 2
 , not 1.6
, not 0.43301…
3
9 3
4
Investigation 1: The Isosceles Right Triangle
Recall: An isosceles triangle has two sides with the same length.
The isosceles right triangle below has one leg of length 1 unit.
P

O
1
Q

Determine the lengths of the other two sides. Leave answers as exact values.

Determine the measure of the two acute angles.

Now imagine putting OPQ on a Cartesian plane with the vertex O at the origin. This
makes  an angle in standard position. What are the coordinates of point P?
Math 20-1
Trigonometry
1. What are the exact values of sin 45 , cos 45 , and tan 45 ? (3 marks)
2. Now imagine reflecting OPQ in the x- and y-axes so that we get three more angles
with the same reference angle as  , one in each quadrant. What are the measures of
the three angles in standard position? (3 marks)
3. Use the reflected triangles to determine the exact value of the sine, cosine and tangent
ratios of each of the three angles above. (3 marks)

How can the CAST rule be used to explain the results of the previous question?
Math 20-1
Trigonometry
Investigation 2: Half an Equilateral Triangle
Recall: An equilateral triangle has all three sides the same length. It also has all three angles the
same. This means that the measure of each angle in an equilateral triangle is ______.
The equilateral triangle below has side length 2 units. Imagine dividing this triangle in half as
shown in the diagram.
P
2
2

2
O
Q

Determine the lengths of each side of OPQ . Leave answers as exact values.

Determine the measure of O and  P .

Now imagine putting OPQ on a Cartesian plane with the vertex O at the origin. This
makes  an angle in standard position. What are the coordinates of point P?
4. What are the exact values of sin 60 , cos 60 , and tan 60 ? (3 marks)
5. Now imagine reflecting OPQ in the x- and y-axes so that we get three more angles
with the same reference angle as  , one in each quadrant. What are the measures of
the three angles in standard position? (3 marks)
Math 20-1
Trigonometry
6. Use the reflected triangles to determine the exact value of the sine, cosine and tangent
ratios of each of the three angles above. (3 marks)
Now imagine flipping the triangle and relabeling it so that the smallest angle is at the origin.
P

O

Q
Imagine putting the new OPQ on a Cartesian plane with the vertex O at the origin. This
makes  an angle in standard position. What are the coordinates of point P?
7. What are the exact values of sin 30 , cos 30 , and tan 30 ? (3 marks)
8. Now imagine reflecting OPQ in the x- and y-axes so that we get three more angles
with the same reference angle as  , one in each quadrant. What are the measures of
the three angles in standard position? (3 marks)
9. Use the reflected triangles to determine the exact value of the sine, cosine and tangent
ratios of each of the three angles above. (3 marks)