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University of Strathclyde Crest Trigonometry Basic Definitions
Introduction & User Info

Section 1:

Laws of Algebra

Section 2:

Exponentials & Logs

Section 3:

Trigonometry

Basic Definitions

Applications Degrees & Radians Pythagoras Theorem Trig Ratios for Acute Angles Trig Ratios for Common Angles Complementary Angles

Angles of Any Size or Sign

Trig Ratios for Angles of any Size or Sign Graphs of Sin, Cos & Tan

Trigonometric Formulae

Pythagorean Identities The Sine and Cosine Rules Addition Formulae Double-Angle and Half-Angle Formulae Solving Trigonometric Equations

Section 4:

Plane Geometry
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The Six Trigonometric Ratios for Acute Angles

You should already have met the trigonometric functions sine (sin θ), cosine (cos θ) and tangent (tan θ).

These functions are often referred to as the trigonometric ratios because they are defined as the ratios of the lengths of the sides of a right-angled triangle.

If θ is one of the acute angles in a right-angled triangle, we can refer to the three sides of the triangle as the adjacent side (A), the opposite side (O) and the hypotenuse (H).

triangle with right angle in bottom-right corner, bottom (A), right-side (O), and diagonal (H) with angle opposite O is θ

The trigonometric ratios of θ are then defined as ratios of the lengths of these sides as follows:

sin θ = O/H; cos θ = A/H; tan θ = O/A

A mnemonic that may help you to remember these ratios is:

SOH - CAH - TOA

The Trigonometric ratios depend only on the angle θ and not on a particular triangle, since all right-angled triangles having an angle θ are similar triangles.

There are three other trigonometric ratios that you may have not yet met, namely: cotangent (cot θ), secant (sec θ) and cosecant (cosec θ).

These are the reciprocals of tan θ, cos θ and sin θ respectively:

cot θ = 1/[tan θ] = A/O; sec theta = 1/[cos θ] = H/A; cosec theta = 1/[sin θ] = H/O

It follows that:

tan theta = [sin θ]/[cos theta]; cot theta = [cos θ]/[sin θ];
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