1. Lesson 3

1.2. Explore

Mathematics 20-1 Module 2

Module 2: Trigonometry

 

Explore

 

In Discover you determined the cosine, sine, and tangent for 0°, 90°, 180°, 270°, and 360° angles using points on a circle with a radius of 2. In Share 1 you may have discussed the fact that the values for these trigonometric ratios are always the same and don’t depend on the radius of the circle.

 

Finding the trigonometric ratios for these angles was straightforward because the terminal arms of each angle landed on either the x- or y-axis, making the point P(x, y) easily known. But what about other angles whose terminal arms do not lie on the x- or y-axis?



glossary

You already saved Module 2 Glossary Terms in your course folder. In this lesson you will define the following terms, and maybe others, in your copy of Module 2 Glossary Terms:

  • 45-45-90 special triangle
  • 30-60-90 special triangle

In the following activities you will investigate special triangles and explore how they can help identify trigonometric ratios for angles that do not lie on the x- or y-axis.

 

Try This 2

 

Follow the steps and answer the questions.

 

Step 1: Consider a square of side length 1 unit.

 

 

This graphic shows a square with side length 1 unit.

 

Step 2: When a diagonal is drawn in the square, what are the angles θ1 and θ2? hint

 

 

This sketch shows a square with side length 1 unit and one diagonal drawn. The diagonal splits the square into two right triangles. The non-90-degree angles in the bottom triangle are labelled theta sub 1 and theta sub 2.

 

Step 3: The diagonal splits the square into two right triangles. What is the length of the hypotenuse? hint

 

 

This sketch shows a square with side length 1 unit with one diagonal drawn and the hypotenuse labelled as h.

 

Step 4: Your diagram should now look like this:

 

 

This graphic shows a square with side length 1 unit and one diagonal drawn. The bottom right triangle has two 45-degree angles. The non-hypotenuse sides are 1 unit, and the hypotenuse is root 2 units.

 

Step 5: Use this triangle to determine cos 45°, sin 45°, and tan 45°. hint

 

course folder Save your responses in your course folder.



This is a right triangle. So, you can use the primary trigonometric ratios you learned in Mathematics 10C (SOH CAH TOA).
The hypotenuse can be determined using the Pythagorean theorem (a2 + b2 = c2).
The diagonal bisects two 90° angles.