Trigonometric Ratios

Why This Matters

Trigonometric ratios are the foundation of everything you'll do with triangles in Geometry—and they show up constantly on exams. You're being tested on your ability to connect angle measures to side lengths in right triangles, which means understanding how sine, cosine, and tangent work as ratios, not just memorizing formulas. These concepts extend into real-world applications like finding heights of buildings, calculating distances, and solving navigation problems.

Here's the key insight: trig ratios are constant for any given angle, regardless of the triangle's size. This principle of similar triangle relationships is what makes trigonometry so powerful. When you see exam questions about angles of elevation, special triangles, or finding missing sides, you're really being asked to demonstrate that you understand these underlying ratio relationships. Don't just memorize SOH CAH TOA—know which ratio to use when and why each formula works.

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The Three Primary Ratios

Every right triangle problem starts here. These ratios compare two specific sides relative to a reference angle (not the right angle). The key is identifying which sides are "opposite," "adjacent," and "hypotenuse" from your angle's perspective.

Sine (sin)

• Opposite over hypotenuse—this ratio measures how the side across from your angle compares to the longest side
• Formula: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$—always produces a value between 0 and 1 for acute angles
• Use sine when you know (or need) the opposite side and hypotenuse—common in height and distance problems

Cosine (cos)

• Adjacent over hypotenuse—this ratio compares the side touching your angle (that isn't the hypotenuse) to the longest side
• Formula: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$—also produces values between 0 and 1 for acute angles
• Cosine and sine are complementary—for any acute angle $\theta$, $\cos(\theta) = \sin(90° - \theta)$

Tangent (tan)

• Opposite over adjacent—this ratio compares the two legs of the right triangle without involving the hypotenuse
• Formula: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$—can produce values greater than 1 (unlike sine and cosine)
• Best choice when the hypotenuse isn't involved—if you know both legs or need to find one leg from the other

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Memory Tools and Relationships

Understanding how the ratios connect helps you work faster and catch errors. These aren't just tricks—they reveal the mathematical structure behind trigonometry.

SOH CAH TOA Mnemonic

• S-O-H: Sine = Opposite/Hypotenuse—the first letter of each word gives you the ratio
• C-A-H: Cosine = Adjacent/Hypotenuse—remember that "adjacent" means next to the angle
• T-O-A: Tangent = Opposite/Adjacent—the only ratio that doesn't use the hypotenuse

Reciprocal Ratios (Cosecant, Secant, Cotangent)

• Cosecant flips sine: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}$—useful when you need hypotenuse in the numerator
• Secant flips cosine: $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}$—less common in basic Geometry but appears in advanced problems
• Cotangent flips tangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}$—sometimes easier than dividing by a tangent value

Pythagorean Theorem Connection

• $a^2 + b^2 = c^2$ underlies all trig ratios—since you're always working with right triangle sides
• Derive missing sides first, then apply ratios—or use trig ratios to find sides when you have an angle
• The identity $\sin^2(\theta) + \cos^2(\theta) = 1$ comes directly from Pythagorean theorem—a key relationship for checking your work

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Special Triangles and Exact Values

These triangles have angle-side relationships you should know without a calculator. Exams love testing whether you recognize these patterns.

45-45-90 Triangle

• Side ratio is $1 : 1 : \sqrt{2}$—the two legs are equal, and the hypotenuse is $\sqrt{2}$ times a leg
• $\sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2}$—these are equal because the triangle is isosceles
• $\tan(45°) = 1$—since opposite and adjacent sides are equal, their ratio is always 1

30-60-90 Triangle

• Side ratio is $1 : \sqrt{3} : 2$—short leg : long leg : hypotenuse
• For 30°: $\sin(30°) = \frac{1}{2}$, $\cos(30°) = \frac{\sqrt{3}}{2}$, $\tan(30°) = \frac{\sqrt{3}}{3}$—the short leg is opposite the smallest angle
• For 60°: values swap—$\sin(60°) = \frac{\sqrt{3}}{2}$, $\cos(60°) = \frac{1}{2}$, $\tan(60°) = \sqrt{3}$

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Real-World Applications

These applications show up constantly in word problems. The setup is everything—draw the triangle and label your angle correctly.

Angle of Elevation and Depression

• Elevation looks UP from horizontal—the angle between your eye level and something above you (top of a building, plane in sky)
• Depression looks DOWN from horizontal—the angle between your eye level and something below you (boat from a cliff, bottom of a valley)
• Both create right triangles with horizontal ground—elevation and depression angles to the same object are actually equal (alternate interior angles)

Right Triangle Relationships

• Trig ratios are constant for any angle—a 30° angle produces the same sine value whether the triangle is 3 inches or 300 feet
• Acute angles in a right triangle are complementary—they sum to 90°, which means $\sin(A) = \cos(B)$ when $A + B = 90°$
• Similar triangles have identical trig ratios—this is why trigonometry works across different-sized triangles

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Extending Beyond Right Triangles

The unit circle bridges basic Geometry trig to more advanced concepts you'll see in later courses.

Unit Circle

• A circle with radius 1 centered at the origin—any point on it has coordinates $(\cos\theta, \sin\theta)$
• Extends trig to all angles—not just acute angles in right triangles, but obtuse, reflex, and negative angles too
• Key points to memorize: $(1, 0)$ at 0°, $(0, 1)$ at 90°, $(-1, 0)$ at 180°, $(0, -1)$ at 270°—these anchor your understanding of how sine and cosine behave

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Quick Reference Table

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Self-Check Questions

1. Which two trig ratios both use the hypotenuse in their formulas, and how do you decide which one to use in a problem?

2. In a 30-60-90 triangle, why does $\sin(30°) = \cos(60°)$? What geometric property explains this?

3. Compare and contrast when you would use the tangent ratio versus the Pythagorean theorem to find a missing side length.

4. If a problem describes an "angle of depression of 25° from the top of a lighthouse," where exactly is that 25° angle located in your diagram, and which trig ratio would you likely use?

5. A right triangle has legs of length 5 and 12. Without using a calculator, explain how you would find $\sin(\theta)$ for the angle opposite the side of length 5. What is the value?