Introduction to Trigonometry — RBSE Class 10 (Mathematics)

Trigonometry literally means "triangle measurement". Stand at the foot of the Qutub Minar, measure the angle up to its top and your distance from it, and trigonometry hands you the height — without ever climbing. This chapter is the dictionary that translates angles into ratios of sides, and back.

1. The six trigonometric ratios

Take a right-angled triangle and focus on one acute angle, call it θ (theta). The three sides get names relative to θ:

• Hypotenuse — the side opposite the right angle (the longest side).
• Opposite (perpendicular) — the side facing θ.
• Adjacent (base) — the remaining side, next to θ.

The six ratios:

$\sin \theta =\frac{\text{opp}}{\text{hyp}},\cos \theta =\frac{\text{adj}}{\text{hyp}},\tan \theta =\frac{\text{opp}}{\text{adj}}$

and their reciprocals:

$\csc \theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{1}{\tan \theta }$

Two relationships fall straight out:

$\tan \theta =\frac{\sin \theta }{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }$

Mnemonic: "Pandit Badri Prasad, Har Har Bhole" → sin = P/H, cos = B/H, tan = P/B (Perpendicular, Base, Hypotenuse). Many students prefer SOH-CAH-TOA.

The ratios depend only on the angle, not on the size of the triangle — all triangles with the same angle θ give the same ratios. That is why a fixed table of values works.

2. The standard-angle table (memorise this)

Memory trick for sin: write 0, 1, 2, 3, 4 under the angles, divide each by 4, then take the square root → $\sqrt{0/4},\sqrt{1/4},\sqrt{2/4},\sqrt{3/4},\sqrt{4/4}$ = 0, 1/2, 1/√2, √3/2, 1. For cos, read the same row backwards. For tan, divide sin by cos.

The reciprocal ratios (cosec, sec, cot) are just 1 ÷ these. Note tan 90° and cot 0° are not defined (division by zero), and so are sec 90° and cosec 0°.

3. The fundamental identities

For any acute angle θ:

$\boxed{{\sin }^2\theta +{\cos }^2\theta =1}$ $1+{\tan }^2\theta ={\sec }^2\theta$ $1+{\cot }^2\theta ={\csc }^2\theta$

The first is just Pythagoras' theorem in disguise (divide $a^2+b^2=c^2$ by $c^2$). The other two come from dividing the first by ${\cos }^2\theta$ and ${\sin }^2\theta$ respectively. These three identities are the workhorses for proving trig identities and for finding one ratio when another is given.

Note: ${\sin }^2\theta$ means $(\sin \theta {)}^2$, not $\sin ({\theta }^2)$.

4. Trigonometric ratios of complementary angles

Two angles are complementary if they add up to 90°. In a right triangle, the two acute angles are complementary, and this swaps the ratios:

$\sin (90°-\theta )=\cos \theta ,\cos (90°-\theta )=\sin \theta$ $\tan (90°-\theta )=\cot \theta ,\cot (90°-\theta )=\tan \theta$ $\sec (90°-\theta )=\csc \theta ,\csc (90°-\theta )=\sec \theta$

So $\sin 60°=\cos 30°$, $\tan 70°=\cot 20°$, and $\frac{\sin 35°}{\cos 55°}=1$ (since $\cos 55°=\sin 35°$). These let you simplify expressions instantly — a favourite 1–2 mark question.

5. Finding all ratios from one

If you're given one ratio, you can find the rest. Example: given $\tan \theta =\frac{4}{3}$, find $\sin \theta$ and $\cos \theta$.

• $\tan \theta =\frac{\text{opp}}{\text{adj}}=\frac{4}{3}$, so opp = 4k, adj = 3k.
• Hypotenuse $=\sqrt{(4k{)}^2+(3k{)}^2}=\sqrt{25k^2}=5k$.
• $\sin \theta =\frac{4k}{5k}=\frac{4}{5}$, $\cos \theta =\frac{3k}{5k}=\frac{3}{5}$.

(Alternatively, use ${\sec }^2\theta =1+{\tan }^2\theta$ to get $\cos \theta$, then $\sin \theta =\cos \theta \cdot \tan \theta$.)

6. Closing thought

This chapter is mostly about fluency, not difficulty. Lock in three things and almost every question opens up:

1. the definitions (SOH-CAH-TOA + the three reciprocals),
2. the standard-angle table, recallable in seconds, and
3. the three identities, with ${\sin }^2\theta +{\cos }^2\theta =1$ as the keystone.

Everything builds on this — the next chapter (Some Applications of Trigonometry) uses these exact ratios to compute heights and distances, and Class 11 extends θ beyond 90° into the full circle. For the RBSE board, expect a "find the value", a "prove the identity" and a "complementary angle" question — all directly from the boxes above.