📐 Level 1 - Topic 5: Trigonometric Ratios in Right Triangles 📐

1) Right Triangles - The Foundation 📐

Trigonometry is deeply connected to triangles, especially right triangles. A right triangle is a triangle with one angle that is exactly \( 90^\circ \) (a right angle).

Key Parts of a Right Triangle:

• Right Angle: The \( 90^\circ \) angle, often marked with a small square.
• Hypotenuse: The side opposite the right angle. It's always the longest side of the right triangle.
• Legs: The two sides that form the right angle. When we consider an acute angle (less than \( 90^\circ \)) within the right triangle, we refer to the legs as:
  • Opposite Side: The leg directly across from the angle we are considering.
  • Adjacent Side: The leg next to the angle we are considering (that is not the hypotenuse).

Consider a right triangle \( \triangle ABC \) where \( \angle C = 90^\circ \). Side \( AB \) is the hypotenuse. If we consider angle \( \angle A \), then side \( BC \) is the opposite side and side \( AC \) is the adjacent side. If we consider angle \( \angle B \), then side \( AC \) is the opposite side and side \( BC \) is the adjacent side.

2) The Six Trigonometric Ratios - Sine, Cosine, Tangent, and Their Reciprocals 🔄

For a right triangle, we define six fundamental trigonometric ratios based on the ratios of the lengths of its sides relative to an acute angle \( \theta \) (let's assume \( \theta \) is one of the non-right angles in the triangle). The three primary ratios are sine, cosine, and tangent.

Mnemonic: SOH CAH TOA

A helpful mnemonic to remember these ratios is SOH CAH TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Say "SOH CAH TOA" to yourself a few times to help memorize these!

Reciprocal Trigonometric Ratios: Cosecant, Secant, Cotangent

There are also three reciprocal trigonometric ratios, which are simply the reciprocals of sine, cosine, and tangent:

• Cosecant of \( \theta \) (csc \( \theta \)): Reciprocal of sine.

  \( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{Hypotenuse}}{\text{Opposite}} \)

• Secant of \( \theta \) (sec \( \theta \)): Reciprocal of cosine.

  \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{Hypotenuse}}{\text{Adjacent}} \)

• Cotangent of \( \theta \) (cot \( \theta \)): Reciprocal of tangent.

  \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Adjacent}}{\text{Opposite}} \)

3) Finding Trigonometric Ratios - Examples 🎯

Example 1: Finding Trigonometric Ratios Given Side Lengths

Example 2: Using Trigonometric Ratios to Find Unknown Side

4) Trigonometric Ratios for Special Angles: \( 30^\circ \), \( 45^\circ \), \( 60^\circ \) 🌟

Certain angles appear very frequently in trigonometry and geometry, and it's essential to know their trigonometric ratios exactly (without relying on calculators). These are \( 30^\circ \), \( 45^\circ \), and \( 60^\circ \) (or \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \) radians).

Trigonometric Ratios for \( 30^\circ \) and \( 60^\circ \) (using a 30-60-90 triangle):

Consider a 30-60-90 right triangle. The ratios of its sides are \( 1 : \sqrt{3} : 2 \) (opposite to \( 30^\circ \) : opposite to \( 60^\circ \) : hypotenuse). Using these ratios:

• \( \sin(30^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{2} \)
• \( \cos(30^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{\sqrt{3}}{2} \)
• \( \tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \) (rationalized)
• \( \sin(60^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{\sqrt{3}}{2} \)
• \( \cos(60^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{2} \)
• \( \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3} \)

Trigonometric Ratios for \( 45^\circ \) (using a 45-45-90 triangle):

Consider a 45-45-90 right triangle (isosceles right triangle). The ratios of its sides are \( 1 : 1 : \sqrt{2} \) (legs : hypotenuse). Using these ratios:

• \( \sin(45^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \) (rationalized)
• \( \cos(45^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \) (rationalized)
• \( \tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{1} = 1 \)

Summary Table of Trigonometric Ratios for Special Angles:

Angle \( \theta \)	\( \sin(\theta) \)	\( \cos(\theta) \)	\( \tan(\theta) \)
\( 30^\circ = \frac{\pi}{6} \) rad	\( \frac{1}{2} \)	\( \frac{\sqrt{3}}{2} \)	\( \frac{\sqrt{3}}{3} \)
\( 45^\circ = \frac{\pi}{4} \) rad	\( \frac{\sqrt{2}}{2} \)	\( \frac{\sqrt{2}}{2} \)	\( 1 \)
\( 60^\circ = \frac{\pi}{3} \) rad	\( \frac{\sqrt{3}}{2} \)	\( \frac{1}{2} \)	\( \sqrt{3} \)

5) Practice Questions 🎯

6) Summary 🎉

• Right Triangles: Triangles with one \( 90^\circ \) angle. Hypotenuse (opposite right angle), legs (opposite & adjacent to acute angles).
• Trigonometric Ratios (SOH CAH TOA):
  • \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
  • \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
  • \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)
• Reciprocal Ratios: Cosecant \( (\csc) \), Secant \( (\sec) \), Cotangent \( (\cot) \) are reciprocals of sine, cosine, tangent respectively.
• Special Angles: Memorize \( \sin, \cos, \tan \) values for \( 30^\circ \), \( 45^\circ \), \( 60^\circ \). Use 30-60-90 and 45-45-90 triangle ratios.
• Applications: Trigonometric ratios relate angles and side lengths in right triangles. Used to find unknown sides or angles.

Congratulations! You've now learned the fundamental trigonometric ratios in right triangles! These ratios are the bedrock for solving many trigonometry problems. Practice identifying opposite, adjacent, hypotenuse, and calculating sine, cosine, tangent for various right triangles. Mastering these ratios will unlock a wide range of applications in trigonometry and beyond! 📐🚀🌟