Right triangle trigonometric ratios Knowledge-Based Study

Study notes

1. Naming Sides Relative to an Acute Angle

   In any right-angled triangle, the three sides are named according to their position relative to one of the acute angles, typically denoted by $\theta$:

   • Hypotenuse: The longest side of the right triangle, directly opposite the right angle ($90^\circ$).
   • Opposite side: The side directly across from the reference angle $\theta$.
   • Adjacent side: The side next to the reference angle $\theta$ that forms the angle along with the hypotenuse.

   Note: If the reference angle changes to the other acute angle, the opposite and adjacent sides switch roles, while the hypotenuse always remains the same.

   Definitions of Sine, Cosine, and Tangent

   Trigonometric ratios express the proportional relationships between pairs of side lengths in a right triangle for a given acute angle $\theta$:

   • Sine ($\sin$): The ratio of the length of the opposite side to the hypotenuse.
     $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
   • Cosine ($\cos$): The ratio of the length of the adjacent side to the hypotenuse.
     $$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$
   • Tangent ($\tan$): The ratio of the length of the opposite side to the adjacent side.
     $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$

   A helpful mnemonic for memorizing these fundamental definitions is SOH-CAH-TOA.

   Definitions of Cosecant, Secant, and Cotangent

   Each primary trigonometric ratio has a corresponding reciprocal ratio defined as follows:

   • Cosecant ($\csc$): The reciprocal of sine.
     $$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$$
   • Secant ($\sec$): The reciprocal of cosine.
     $$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$$
   • Cotangent ($\cot$): The reciprocal of tangent.
     $$\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}}$$

   Notice also that the tangent and cotangent ratios can be expressed directly in terms of sine and cosine:
   $$\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}$$

   Finding All Six Ratios from Given Sides

   Consider a right triangle with legs of lengths $a = 3$ and $b = 4$. Let $\theta$ be the acute angle opposite the side of length 3.

   Step 1: Determine the Hypotenuse

   Using the Pythagorean theorem:
   $$c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

   Step 2: Identify Side Designations Relative to $\theta$

   • $\text{Opposite} = 3$
   • $\text{Adjacent} = 4$
   • $\text{Hypotenuse} = 5$

   Step 3: Calculate the Ratios

   • $\sin \theta = \frac{3}{5}$ and $\csc \theta = \frac{5}{3}$
   • $\cos \theta = \frac{4}{5}$ and $\sec \theta = \frac{5}{4}$
   • $\tan \theta = \frac{3}{4}$ and $\cot \theta = \frac{4}{3}$

   Ratios for $30^\circ$, $45^\circ$, and $60^\circ$

   Using standard geometric constructions (an equilateral triangle bisected into two $30^\circ\text{--}60^\circ\text{--}90^\circ$ triangles, and an isosceles $45^\circ\text{--}45^\circ\text{--}90^\circ$ right triangle), we obtain exact values:

   • For $30^\circ$: $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\tan 30^\circ = \frac{\sqrt{3}}{3}$
   • For $45^\circ$: $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\tan 45^\circ = 1$
   • For $60^\circ$: $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, $\tan 60^\circ = \sqrt{3}$

   Key Takeaway

   For any two complementary acute angles $A$ and $B$ where $A + B = 90^\circ$, cofunction relationships hold:
   $$\sin A = \cos(90^\circ - A) = \cos B$$