Trigonometric Ratios

Why This Matters

Trigonometric ratios connect angle measures to side lengths in right triangles. They show up constantly in Geometry exams and extend into real-world problems like finding the height of a building or calculating a distance you can't measure directly.

The core principle: trig ratios are constant for any given angle, no matter the triangle's size. A 30° angle produces the same sine value whether the triangle is 3 inches tall or 300 feet tall. This works because of similar triangle relationships. So don't just memorize SOH CAH TOA. Focus on which ratio to use when and why each one 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 (never the right angle itself). The key is identifying which sides are "opposite," "adjacent," and "hypotenuse" from your angle's perspective.

• Opposite: the side directly across from your angle
• Adjacent: the side touching your angle that is not the hypotenuse
• Hypotenuse: always the longest side, always across from the 90° angle

Sine (sin)

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

This ratio measures how the side across from your angle compares to the longest side. For acute angles, sine always produces a value between 0 and 1. Use sine when you know (or need) the opposite side and hypotenuse.

Cosine (cos)

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

This ratio compares the side touching your angle (that isn't the hypotenuse) to the longest side. Also produces values between 0 and 1 for acute angles. Cosine and sine are linked by a complementary relationship: for any acute angle $\theta$, $\cos(\theta) = \sin(90° - \theta)$.

Tangent (tan)

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

This ratio compares the two legs of the right triangle without involving the hypotenuse at all. Unlike sine and cosine, tangent can produce values greater than 1. Use tangent when the hypotenuse isn't part of the problem.

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Memory Tools and Relationships

Understanding how the ratios connect helps you work faster and catch errors. These relationships reveal the mathematical structure behind trigonometry.

SOH CAH TOA Mnemonic

• S-O-H: Sine = Opposite / Hypotenuse
• C-A-H: Cosine = Adjacent / Hypotenuse
• T-O-A: Tangent = Opposite / Adjacent

The first letter of each word gives you the ratio. "Adjacent" means the leg next to your reference angle.

Reciprocal Ratios (Cosecant, Secant, Cotangent)

Each primary ratio has a reciprocal that flips the fraction:

• Cosecant flips sine: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}$
• Secant flips cosine: $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}$
• Cotangent flips tangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}$

These appear less often in standard Geometry, but you should recognize them. A common naming trap: cosecant is the reciprocal of sine (not cosine), and secant is the reciprocal of cosine (not sine). The names cross over.

Pythagorean Theorem Connection

Since trig ratios always involve right triangle sides, the Pythagorean theorem ($a^2 + b^2 = c^2$) is never far away. You can:

• Derive a missing side with the Pythagorean theorem, then apply a trig ratio
• Or use a trig ratio to find a side when you have an angle and one side

The identity $\sin^2(\theta) + \cos^2(\theta) = 1$ comes directly from dividing the Pythagorean theorem by $c^2$. It's useful for checking your work: if you've found both sine and cosine for an angle, their squares should add to 1.

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Special Triangles and Exact Values

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

45-45-90 Triangle

Side ratio: $1 : 1 : \sqrt{2}$

The two legs are equal (it's an isosceles right triangle), and the hypotenuse is $\sqrt{2}$ times a leg.

• $\sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2}$ (equal because the legs are equal)
• $\tan(45°) = 1$ (opposite and adjacent are the same length, so their ratio is always 1)

30-60-90 Triangle

Side ratio: $1 : \sqrt{3} : 2$ (short leg : long leg : hypotenuse)

The short leg is opposite the 30° angle, the long leg is opposite the 60° angle, and the hypotenuse is opposite the 90° angle.

For 30°:

• $\sin(30°) = \frac{1}{2}$, $\cos(30°) = \frac{\sqrt{3}}{2}$, $\tan(30°) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

For 60° the sine and cosine values swap:

• $\sin(60°) = \frac{\sqrt{3}}{2}$, $\cos(60°) = \frac{1}{2}$, $\tan(60°) = \sqrt{3}$

Notice that $\sin(30°) = \cos(60°)$ and $\sin(60°) = \cos(30°)$. This is the complementary relationship in action: 30° and 60° sum to 90°.

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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 before writing any equation.

Angle of Elevation and Depression

• Elevation looks UP from horizontal: the angle between your eye level and something above you (top of a building, a plane in the sky)
• Depression looks DOWN from horizontal: the angle between your eye level and something below you (a boat seen from a cliff, the bottom of a valley)

Both create right triangles with a horizontal line. An important geometric fact: the angle of elevation from point A to point B equals the angle of depression from point B to point A (alternate interior angles with a horizontal transversal).

Solving a Word Problem: Step-by-Step

1. Draw the right triangle and label the known angle (not the 90° angle)
2. Mark the horizontal and vertical sides
3. Identify which sides are opposite, adjacent, and hypotenuse relative to your labeled angle
4. Determine which sides are known and which you need to find
5. Pick the trig ratio that connects those two sides (SOH CAH TOA)
6. Set up the equation and solve for the unknown

Right Triangle Relationships

• Trig ratios are constant for any given angle: scaling the triangle up or down doesn't change the ratios
• The two acute angles in a right triangle are complementary: they sum to 90°, so $\sin(A) = \cos(B)$ 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 Geometry trig to more advanced concepts you'll encounter in later courses.

Unit Circle

A circle with radius 1 centered at the origin. Any point on it has coordinates $(\cos\theta, \sin\theta)$, which means the x-coordinate gives you cosine and the y-coordinate gives you sine.

This extends trig beyond acute angles to all angles, including obtuse, reflex, and negative angles.

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 as the angle rotates around the circle.

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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 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, find $\sin(\theta)$ for the angle opposite the side of length 5. (Hint: you'll need to find the hypotenuse first.)