Trigonometric ratios in right triangle — sin, cos, tan, cosec, sec, cot

The Setup: What is a Right Triangle?

A right triangle has:

• One 90° angle (the right angle)
• Two other acute angles (less than 90°)
• Three sides: the hypotenuse (longest, opposite the right angle), and two legs

Pick one of the acute angles and call it θ (theta). Now the sides get names relative to θ:

• Opposite side: the side across from θ
• Adjacent side: the side next to θ (not the hypotenuse)
• Hypotenuse: always the longest side

WHY these names? Because the ratios change meaning if you switch which angle you're measuring. The "opposite" for angle A becomes "adjacent" for angle B.

The Six Trigonometric Ratios

$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

$\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} \quad \sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} \quad \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}}$

Notice: cosec, sec, cot are just reciprocals of sin, cos, tan.

WHY six ratios, not three? Historically, before calculators, having all six made certain calculations faster. Today, we mostly use sin, cos, tan; the others are convenient shortcuts.

HOW to remember? Use SOH-CAH-TOA:

• Sin = Oposite/Hypotenuse
• Cos = Adjacent/Hypotenuse
• Tan = Opposite/Adjacent

Derivation from First Principles

WHERE do these come from?

Start with similar triangles. If two right triangles have the same angle θ, they are similar (same shape, different size). In similar triangles, ratios of corresponding sides are equal.

Consider two right triangles with angle θ:

• Triangle 1: opposite = a, adjacent = b, hypotenuse = c
• Triangle 2: opposite = ka, adjacent = kb, hypotenuse = kc (scaled by factor k)

The ratio opposite/hypotenuse: $\frac{a}{c} = \frac{ka}{kc} = \frac{a}{c}$

The ratio stays the same! This ratio depends only on θ, not the triangle's size. We define this ratio as sin θ.

WHY is this useful? Once we measure sin θ for one triangle, it works for ALL triangles with that angle. That's the power: trigonometric ratios are properties of the angle, not the triangle.

Derivation: $\sin \theta = \frac{o}{h} \implies \csc \theta = \frac{h}{o} = \frac{1}{\sin \theta}$

Similarly for sec and cot. WHY introduce these? They make certain identities and integrals cleaner.

Derivation: $\tan \theta = \frac{o}{a}, \quad \sin \theta = \frac{o}{h}, \quad \cos \theta = \frac{a}{h}$

Divide sin by cos: $\frac{\sin \theta}{\cos \theta} = \frac{o/h}{a/h} = \frac{o}{h} \cdot \frac{h}{a} = \frac{o}{a} = \tan \theta$

WHY this matters: If you know sin and cos, you can always find tan without measuring the triangle again.

Worked Examples

Solution: First, find hypotenuse using Pythagorean theorem: $h^2 = o^2 + a^2= 3^2 + 4^2 = 9 + 16 = 25 \implies h = 5$

Why Pythagorean theorem? Every right triangle obeys $a^2 + b^2 = c^2$. This comes from the geometry of squares built on each side.

Now calculate: $\sin \theta = \frac{3}{5} = 0.6$ $\cos \theta = \frac{4}{5} = 0.8$ $\tan \theta = \frac{3}{4} = 0.75$

$\csc \theta = \frac{5}{3} \approx 1.667$ $\sec \theta = \frac{5}{4} = 1.25$ $\cot \theta = \frac{4}{3} \approx 1.333$

Verification: Check that $\sin^2 \theta + \cos^2 \theta = (0.6)^2 + (0.8)^2 = 0.36 + 0.64 = 1$ ✓

Why this step? The Pythagorean identity is a built-in consistency check for right triangles.

Solution: $\sin \theta = \frac{o}{h} = \frac{5}{13}$

Given $h = 26$: $\frac{o}{26} = \frac{5}{13} \implies o = 26 \cdot \frac{5}{13} = 10 \text{ cm}$

Why this step? We're using the definition of sine and cross-multiplying.

For adjacent, use Pythagorean theorem: $a^2 = h^2 - o^2 = 26^2 - 10^2 = 676 - 100 = 576 \implies a = 24 \text{ cm}$

Check: $\cos \theta = \frac{24}{26} = \frac{12}{13}$, and $\sin^2 \theta + \cos^2 \theta = \left(\frac{5}{13}\right)^2 + \left(\frac{12}{13}\right)^2 = \frac{25 + 144}{169} = \frac{169}{169} = 1$ ✓

Solution: $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{7/25}{24/25} = \frac{7}{25} \cdot \frac{25}{24} = \frac{7}{24}$

Why this works? The quotient identity lets us compute tan from sin and cos algebraically, no geometry needed. This becomes crucial when working with abstract angles beyond right triangles.

Common Mistakes & How to Fix Them

Why it feels right: "Adjacent" in English means "next to," and the hypotenuse is indeed next to the angle.

The fix: Adjacent means "the leg that forms one side of the angle θ" (not the hypotenuse). The hypotenuse is ALWAYS opposite the right angle, never called adjacent. Draw the triangle, label θ clearly, and trace which side is across from θ (opposite) and which side θ "sits on" (adjacent).

Steel-man the mistake: The confusion arises because we're using "adjacent" in a technical sense. Always remember: in trig, "adjacent" and "opposite" are relative to the angle θ, and hypotenuse is its own category.

Why it feels right: The names sound similar, and students confuse "cosecant" with "cosine secant."

The fix: Cosecant is the RECIPROCAL of sine: $\csc \theta = \frac{1}{\sin \theta} = \frac{h}{o}$. Memorize: csc flips sin, sec flips cos, cot flips tan.

Mnemonic: "Cosecant Cuts Sine" (flips it upside down).

Why it feels right: Students forget that opposite/adjacent are angle-dependent.

The fix: Label your angle first, then identify opposite/adjacent relative to that specific angle. If you switch angles, the ratios change. For angle A, opposite is different than opposite for angle B.

Why These Ratios Are Fundamental

Physical interpretation:

• sin θ tells you the vertical component of a unit vector at angle θ
• cos θ tells you the horizontal component
• tan θ is the slope of the line making angle θ with the horizontal

Applications:

• Navigation: Breaking velocity/force into components
• Waves: sin and cos describe oscillations (sound, light, springs)
• Engineering: Calculating tensions, angles of incline, bridge supports

These six ratios are the foundation for all of trigonometry, from solving triangles to Fourier series.

The cool part? Once you know the angle, you can figure out ALL of this without measuring anything else. That's why these ratios are so powerful—they turn angles into distances and distances into angles.

Connections

• Pythagoras theorem — Used to find the third side
• Similar triangles — Why ratios depend only on angle
• Unit circle definition of trig functions — Extends these ratios beyond right triangles
• Trigonometric identities — Pythagorean, quotient, reciprocal
• Solving right triangles — Practical applications
• Components of vectors — sin/cos give x, y components
• Angle of elevation and depression — Real-world problems

#flashcards/maths

What is the definition of sin θ in a right triangle? :: sin θ = opposite/hypotenuse

What is the definition of tan θ in a right triangle? :: tan θ = opposite/adjacent

If cos θ = 0.8 and sin θ = 0.6, find tan θ without the triangle :: tan θ = sin θ / cos θ = 0.6/0.8 = 0.75

Concept Map

Hinglish (regional understanding)