Trigonometric Ratios

Trigonometry connects angles to side lengths in right-angled triangles. It is one of the most useful topics in mathematics -- used in engineering, navigation, architecture, and even music.

The Three Basic Ratios: SOH-CAH-TOA

In a right-angled triangle, relative to an angle theta:

sin(theta) = Opposite / Hypotenuse (SOH)

cos(theta) = Adjacent / Hypotenuse (CAH)

tan(theta) = Opposite / Adjacent (TOA)

The hypotenuse is always the longest side, opposite the right angle. The opposite side is across from the angle. The adjacent side is next to the angle (not the hypotenuse).

Standard Angle Values

You should memorize these -- they appear constantly in Grade 11 math.

| Angle | sin | cos | tan | |-------|-----|-----|-----| | 0 degrees | 0 | 1 | 0 | | 30 degrees | 1/2 | root3/2 | 1/root3 | | 45 degrees | 1/root2 | 1/root2 | 1 | | 60 degrees | root3/2 | 1/2 | root3 | | 90 degrees | 1 | 0 | undefined |

Memory Trick: For sin values of 0, 30, 45, 60, 90 degrees, think: root0/2, root1/2, root2/2, root3/2, root4/2. That gives you 0, 1/2, 1/root2, root3/2, 1.

Worked Example

In a right triangle, the side opposite to angle A is 3 cm and the hypotenuse is 5 cm. Find sin(A), cos(A), and tan(A).

First, find the adjacent side: adjacent = root(25 - 9) = root(16) = 4 cm

• sin(A) = 3/5 = 0.6
• cos(A) = 4/5 = 0.8
• tan(A) = 3/4 = 0.75

Key Identity

sin^2(theta) + cos^2(theta) = 1 -- This is true for ALL angles and is extremely useful for simplifying expressions.

Nepal Connection: When surveyors measured the height of Sagarmatha (Mount Everest), they used trigonometric ratios and angles measured from known distances. The original measurement in 1856 relied heavily on trigonometry!

Key Takeaways

• SOH-CAH-TOA helps you remember which ratio uses which sides
• Memorize the values for standard angles (0, 30, 45, 60, 90 degrees)
• sin^2 + cos^2 = 1 is the most important trigonometric identity
• Always identify opposite, adjacent, and hypotenuse relative to the given angle