Exact Values of Trig Ratios & CAST

Find exact values and learn CAST rule for all quadrants. Grade 11 Math 20-1 Alberta.

Lesson 4.3 of Trigonometry in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Trig Ratios on the Coordinate Plane. Redefine sine, cosine, and tangent using x, y, and r coordinates instead of triangle sides. Work through examples where a point on the terminal arm gives exact trig ratios.
• Signs of the Ratios and the CAST Rule. Learn how the sign of each trig ratio depends on the quadrant, and use the CAST rule to quickly determine which ratios are positive in each quadrant.
• Finding Exact Ratios from a Given Ratio. Given one trig ratio and a quadrant, find the other two exact trig ratios without a calculator. Practice rationalizing denominators and applying the CAST rule.

Inside the lesson — a free preview

From Triangles to Coordinates

A New Way to Think About Trig. When we worked with right triangles, we defined the trig ratios in terms of opposite, adjacent, and hypotenuse. Now we will redefine them in terms of x, y, and r — coordinates on a coordinate plane.

Suppose θ is any angle in standard position, and P(x, y) is any point on the terminal arm. Let r be the distance from the origin to point P.

Connecting x, y, and r

The Pythagorean Relationship. The point P, the origin, and the point directly below P on the x-axis form a right triangle. So x, y, and r are related by the Pythagorean theorem:

x^2 + y^2 = r^2

Solving for r:

r = √(x^2 + y^2)

Key Fact: r is Always Positive. Since r represents a distance from the origin, r is never negative. But x and y can each be positive or negative, depending on which quadrant P lies in.

Redefining the Trig Ratios

From SOH CAH TOA to x, y, r. Looking at the right triangle formed by the terminal arm:

The opposite side (relative to θ) is the vertical leg, with length y.

The adjacent side (relative to θ) is the horizontal leg, with length x.

The hypotenuse is the terminal arm, with length r.

So we can rewrite SOH CAH TOA:

θ = (y)/(r)

θ = (x)/(r)

θ = (y)/(x)

Why This Matters. These definitions work for any angle in standard position, not just acute angles in right triangles. This is the key upgrade — trig now applies to obtuse angles, reflex angles, and beyond.

Worked example: Find the Exact Trig Ratios

Example 1 — Point P(-8, 15) on the Terminal Arm. The point P(-8, 15) lies on the terminal arm of an angle θ in standard position. Determine the exact trigonometric ratios for θ, θ, and θ.

Step 1 — Identify x and y.

x = -8, y = 15

Step 2 — Find r using the Pythagorean theorem.

r = √(x^2 + y^2) = √((-8)^2 + 15^2) = √(64 + 225) = √(289) = 17

Note: r is positive, even though x is negative.

Step 3 — Apply the definitions.

θ = (y)/(r) = (15)/(17)

θ = (x)/(r) = (-8)/(17)

θ = (y)/(x) = (15)/(-8) = -(15)/(8)

Answer: θ = (15)/(17), θ = -(8)/(17), θ = -(15)/(8)

Notice the Signs. Two of the three ratios are negative. That happened because x is negative — and x appears in θ and θ. The sign of each ratio depends on the quadrant.

Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.