Trigonometric Ratios

Understand sine, cosine, tangent, and reciprocal ratios. Math 30-1 Alberta.

Lesson 4.3 of Trigonometric Ratios & Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Six Trigonometric Ratios. Learn the six fundamental trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. Master SOH CAH TOA and discover how these ratios connect to coordinates on the unit circle.
• Signs of Trigonometric Ratios (CAST Rule). Understand how the signs of trig ratios depend on the quadrant. Master the CAST rule mnemonic and learn to evaluate quadrantal angles at 0°, 90°, 180°, and 270°.
• Finding Exact Trig Values. Use the unit circle to find exact trigonometric values. Learn how coordinates (x, y) on the unit circle give you cos θ, sin θ, and tan θ for any angle.

Inside the lesson — a free preview

Trigonometric Ratios

Trigonometric ratios describe relationships between the sides of a right triangle relative to an angle θ.

Primary Trigonometric Ratios. θ = (opposite)/(hypotenuse)

θ = (adjacent)/(hypotenuse)

θ = (opposite)/(adjacent)

Memory Trick: SOH CAH TOA

A common memory trick to remember the trigonometric ratios is:

SOH CAH TOA

SOH → = (Opposite)/(Hypotenuse)

CAH → = (Adjacent)/(Hypotenuse)

TOA → = (Opposite)/(Adjacent)

Reciprocal Ratios

Each primary trig ratio has a reciprocal:

Reciprocal Trigonometric Ratios. θ = (hypotenuse)/(opposite) (cosecant)

θ = (hypotenuse)/(adjacent) (secant)

θ = (adjacent)/(opposite) (cotangent)

As Reciprocals. These can also be written as:

θ = (1)/(θ)

θ = (1)/(θ)

θ = (1)/(θ)

Writing Tangent and Cotangent Using Other Ratios

Tangent and cotangent can also be written as quotients of sine and cosine:

θ = (θ)/(θ)

θ = (θ)/(θ)

Trigonometric Ratios Using Coordinates

Suppose the terminal arm of angle θ passes through a point:

P(x, y)

Let:

r = √(x^2 + y^2)

where r is the distance from the origin to point P.

Unit Circle Connection

On the Unit Circle, the radius is:

r = 1

So the coordinates of a point on the circle can be written as:

(x, y) = (θ, θ)

Signs of Trigonometric Ratios

The signs of trigonometric ratios depend on the quadrant in which the angle lies.

Since we know that on the unit circle:

• θ = y

• θ = x

• θ = (y)/(x)

The signs of x and y in each quadrant determine which ratios are positive or negative.

Worked example: Evaluating Quadrantal Angles

Example 1: Find 90°

Solution: At 90°, the coordinate is (0, 1)

Since θ = y, we have:

90° = 1

Example 2: Find 180°

Solution: At 180°, the coordinate is (-1, 0)

Since θ = x, we have:

180° = -1

Example 3: Find 0°

Solution: At 0°, the coordinate is (1, 0)

Since θ = (y)/(x), we have:

0° = (0)/(1) = 0

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