Right-Triangle Trigonometry & Special Triangles

Fundamental Right-Triangle Concepts

• A right triangle contains one $90^\circ$ angle.

• Side names relative to an acute angle $\theta$:

  • Hypotenuse (c): side opposite the right angle; always the longest.

  • Opposite (o): side opposite the chosen angle $\theta$.

  • Adjacent (a): side next to $\theta$ that is not the hypotenuse.

• Classic theorem:
  $a^2 + b^2 = c^2$ (Pythagorean Theorem).

• Pythagorean triples (integer solutions): e.g. $3-4-5$, $5-12-13$, $6-8-10$.

  • Historical note: Pythagoreans zealously guarded discoveries of new triples ("first mathematical cult").

Finding Missing Sides (examples)

• Example 1: sides $5$ and $12$, find hypotenuse.

  • $25 + 144 = c^2 \Rightarrow c = 13$.

• Example 2: sides $10$ (hyp) and $4$ (leg), find other leg.

  • $a^2 + 4^2 = 10^2 \Rightarrow a = \sqrt{84}=2\sqrt{21}$.

• Example 3: legs $2,3$ → $c = \sqrt{13}$ (simplify radicals only; do not decimal-approximate in exact answers).

Right-Triangle Definitions of Trig Ratios

• Using $\theta$ in a right triangle:

  • $\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$

  • $\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}$

  • $\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}$

  • Reciprocals:

  • $\csc\theta = \dfrac{1}{\sin\theta}=\dfrac{\text{Hypotenuse}}{\text{Opposite}}$

  • $\sec\theta = \dfrac{1}{\cos\theta}=\dfrac{\text{Hypotenuse}}{\text{Adjacent}}$

  • $\cot\theta = \dfrac{1}{\tan\theta}=\dfrac{\text{Adjacent}}{\text{Opposite}}$

• Keep two "tattooed" reference systems: one arm = unit-circle definitions, other arm = right-triangle definitions.

Unit-Circle vs. Right-Triangle View

• Unit circle (radius $1$) point $(x,y)$ with same $\theta$:

  • $\sin\theta = y$, $\cos\theta = x$, $\tan\theta = \dfrac{y}{x}$.

• Build a right triangle by dropping a perpendicular → leg lengths $x$ and $y$, hypotenuse $1$.

  • Matches right-triangle ratios exactly; thus both definitions are equivalent.

Reciprocal & Quotient Identities (review)

• Reciprocal pairs all satisfy $f(\theta)\cdot f^{-1}(\theta)=1$:

  • $\sin\theta\,\csc\theta = 1$, $\cos\theta\,\sec\theta = 1$, $\tan\theta\,\cot\theta = 1$.

• Quotient:

  • $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$.

Pythagorean Identities (derived from $a^2+b^2=c^2$)

• $\sin^2\theta + \cos^2\theta = 1$

• $1 + \tan^2\theta = \sec^2\theta$

• $1 + \cot^2\theta = \csc^2\theta$

Worked Example: Finding Remaining Ratios from Two

Given $\sin\theta = \dfrac{2}{3},\; \cos\theta = \dfrac{\sqrt{5}}{3}$.

• Reciprocal quickly: $\csc\theta=\dfrac{3}{2},\; \sec\theta=\dfrac{3\sqrt{5}}{5}$.

• $\tan\theta = \dfrac{\sin}{\cos}=\dfrac{2/3}{\sqrt{5}/3}=\dfrac{2}{\sqrt{5}}=\dfrac{2\sqrt{5}}{5}$.

• $\cot\theta = \dfrac{1}{\tan\theta}=\dfrac{\sqrt{5}}{2}$.

Special Right Triangles

30-60-90 Triangle

• Start with equilateral $\triangle$ (all sides $2$). Bisect to form:

  • Hypotenuse $=2$

  • Short leg (opposite $30^\circ$) $=1$

  • Long leg (opposite $60^\circ$) $=\sqrt{3}$

• Key ratios:

  • $\sin30^\circ = \tfrac12$

  • $\cos30^\circ = \dfrac{\sqrt{3}}{2}$

  • $\tan30^\circ = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$

  • $\sin60^\circ = \dfrac{\sqrt{3}}{2}$

  • $\cos60^\circ = \tfrac12$

  • $\tan60^\circ = \sqrt{3}$

• For any 30-60-90 triangle: hypotenuse $=2k$, short leg $=k$, long leg $=k\sqrt{3}$.

45-45-90 Triangle

• Isosceles right triangle; let legs $=a$.

  • $a^2+a^2=2^2\Rightarrow a=\sqrt{2}$ when hypotenuse set to $2$.

• Ratios (same for both acute angles):

  • $\sin45^\circ = \cos45^\circ = \dfrac{\sqrt{2}}{2}$

  • $\tan45^\circ = 1$

  • $\csc45^\circ = \sec45^\circ = \sqrt{2}$

  • $\cot45^\circ = 1$

Rationalizing Denominators

• If a radical appears in a denominator, multiply by an appropriate $1$ (e.g. $\dfrac{\sqrt{n}}{\sqrt{n}}$) to move the radical to the numerator.

  • Example: $\dfrac{1}{\sqrt{5}}\times\dfrac{\sqrt{5}}{\sqrt{5}}=\dfrac{\sqrt{5}}{5}$.

Calculator Usage Tips

• Always set MODE correctly:

  • For triangles, switch from default Radians to Degrees.

• Keys: SIN, COS, TAN expect degree input when in degree mode.

• Calculators usually output decimal approximations. Recognize common special-angle decimals:

  • $\sin30^\circ=0.5$

  • $\cos30^\circ\approx0.8660254\ldots (\sqrt{3}/2)$

  • $\cos45^\circ\approx0.7071067\ldots (\sqrt{2}/2)$

• Double-check odd answers by verifying largest side ↔ largest angle, etc.

Solving Right Triangles (General Procedure)

1. Sketch and label: right angle at $C$, other angles $A,B$; opposite sides $a,b$; hypotenuse $c$.

2. Easiest first: find missing acute angle using $A+B=90^\circ$.

3. Prefer given data: when possible use sides/angles provided in the problem instead of values you computed (reduces rounding errors).

4. Choose ratio matching the unknown:

   • Need hypotenuse? Probably use $\sin$ or $\cos$.

   • Need opposite or adjacent? Pick among $\sin,\cos,\tan$ accordingly.

5. Solve algebraically, then compute with calculator (degree mode).

6. Check: largest angle ↔ largest side; smallest angle ↔ smallest side; $a^2+b^2\approx c^2$.

Example Walk-Through 1

Given $B=69^\circ,\; b=37$.

• $A = 90-69 = 21^\circ$.

• Solve for $c$ using $\cos$ (adjacent/hyp): $\cos69^\circ = 37/c \Rightarrow c = 37/\cos69^\circ \approx 103.2$.

• Solve for $a$ using $\tan$ (opp/adj): $\tan69^\circ = a/37 \Rightarrow a = 37\tan69^\circ \approx 100.4$.

Example Walk-Through 2

Given $B=57.4^\circ,\; b=70$.

• $A = 32.6^\circ$.

• $c = \dfrac{70}{\sin57.4^\circ} \approx 83.1$.

• $a = \dfrac{70}{\tan57.4^\circ} \approx 44.8$.

Example Walk-Through 3

Given $c = 343.6,\; B = 32.17^\circ$.

• $A = 57.83^\circ$.

• $a = c\cos B = 343.6\cos32.17^\circ \approx 290.8$.

• $b = c\sin B = 343.6\sin32.17^\circ \approx 182.9$.

Cofunction Identities

• Definition: Trig functions paired with a "co-" version (sine ↔ cosine, tangent ↔ cotangent, secant ↔ cosecant).

• Rule: Cofunctions of complementary angles are equal.

  • If $A + B = 90^\circ$,

  • $\sin A = \cos B$,

  • $\tan A = \cot B$,

  • $\sec A = \csc B$.

• Example: $\sin30^\circ = \cos60^\circ = 0.5$; $\sin57.83^\circ \approx \cos32.17^\circ$.

Key Study Tips

• Memorize exactly: all six trig ratios, reciprocal pairs, quotient rules.

• Know by heart values for $30^\circ, 45^\circ, 60^\circ$ (both sine & cosine—and thus tangent).

• Practice rationalizing denominators quickly.

• Check calculator mode before every computation.

• Label diagrams consistently; avoid mixing derived numbers if an original value is available.

• For exam #2 (not trig): focus review on quadratic → logarithmic functions; trig resumes afterward.