Special Right Triangles and Trigonometric Ratios

Special Right Triangle Properties

  • 30-60-90 Right Triangles     - In a 30exto−60exto−90exto30^ ext{o}-60^ ext{o}-90^ ext{o} triangle, the sides follow specific ratios based on the short leg.     - Shorter Leg: Categorized as xx.     - Hypotenuse: Defined as 2x2x (equivalent to 2×shorter leg2 \times \text{shorter leg}).     - Longer Leg: Defined as x3x\sqrt{3} (equivalent to shorter leg⋅3\text{shorter leg} \cdot \sqrt{3}).     - Note the location of the angles: the shorter leg is opposite the 30exto30^ ext{o} angle, and the longer leg is opposite the 60exto60^ ext{o} angle.
  • 45-45-90 Right Triangles     - The transcript lists for the 45exto45^ ext{o} triangle:         - hypotenuse=leg\text{hypotenuse} = \text{leg}

Introduction to Trigonometry

  • Usage and Purpose     - Trigonometry is used to find the lengths of sides in a right triangle in instances where the Pythagorean Theorem or Special Right Triangles rules are not applicable or won't work.
  • Component Definitions     - Opposite: The side across from the designated angle θ\theta.     - Adjacent: The side next to the designated angle θ\theta that is not the hypotenuse.     - Hypotenuse: The longest side of the right triangle, located across from the right angle.

Trigonometric Ratios (SOH CAH TOA)

  • Sine (SOH)     - Formula: sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}     - The mnemonic SOH indicates Sine is Opposite over Hypotenuse.
  • Cosine (CAH)     - Formula: cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}     - The mnemonic CAH indicates Cosine is Adjacent over Hypotenuse.
  • Tangent (TOA)     - Formula: tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}     - The mnemonic TOA indicates Tangent is Opposite over Adjacent.