Trigonometry – Basics and Ratios

🔹 What is Trigonometry?

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, especially right-angled triangles.

It helps solve problems related to heights, distances, and angles.

🔹 Right-Angled Triangle Basics

A triangle with one angle equal to 90°.
Sides are named relative to the angle of interest:
- Hypotenuse (H): The longest side opposite the right angle.
- Opposite side (O): Side opposite to the angle considered.
- Adjacent side (A): Side next to the angle considered (other than hypotenuse).

🔹Trigonometric Ratios

For an angle θ in a right triangle:

Ratio	Formula	Meaning
Sin θ	$\frac{O}{H}$	Sine = Opposite / Hypotenuse
Cos θ	$\frac{A}{H}$	Cosine = Adjacent / Hypotenuse
Tan θ	$\frac{O}{A}$	Tangent = Opposite / Adjacent
Cosec θ	$\frac{H}{O}$	Cosecant = Hypotenuse / Opposite
Sec θ	$\frac{H}{A}$	Secant = Hypotenuse / Adjacent
Cot θ	$\frac{A}{O}$	Cotangent = Adjacent / Opposite

🔹Trigonometry Table

θ (Degrees)	θ (Radians)	sin θ	cos θ	tan θ	cot θ	sec θ	cosec θ
0°	0	0	1	0	∞	1	∞
30°	π/6	1/2	√3/2	1/√3	√3	2/√3	2
45°	π/4	1/√2	1/√2	1	1	√2	√2
60°	π/3	√3/2	1/2	√3	1/√3	2	2/√3
90°	π/2	1	0	∞	0	∞	1
180°	π	0	–1	0	∞	–1	∞
270°	3π/2	–1	0	∞	0	∞	–1

🔹Important Trigonometric Identities

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

🔹 Common Angle Values (in degrees and radians)

θ (degrees)	0°	30°	45°	60°	90°
Sin θ	0	1/2	√2/2	√3/2	1
Cos θ	1	√3/2	√2/2	1/2	0
Tan θ	0	√3/3	1	√3	∞

🔹Using Trigonometry to Solve Problems

Use Pythagoras theorem to find missing sides:
$H^2 = O^2 + A^2$
Use trigonometric ratios to find unknown sides or angles.
Remember the angle of elevation and depression concepts:
- Elevation: angle above the horizontal line.
- Depression: angle below the horizontal line.

🔹 Examples

Example 1:
In a right triangle, if the hypotenuse is 10 cm and the angle θ = 30°, find the opposite side.

Solution:
$\sin 30° = \frac{O}{10}$
$1/2 = \frac{O}{10} \Rightarrow O = 5 \text{ cm}$

Example 2:

If tan θ = 1, find sin θ and cos θ.

✅ Solution:

Given:
tan θ = 1 ⇒ Opposite side = Adjacent side

Let both the opposite and adjacent sides be 1 unit.

Then using the Pythagoras Theorem:

Hypotenuse = √(Opposite² + Adjacent²)
= √(1² + 1²)
= √2

Now calculate:

sin θ = Opposite / Hypotenuse = 1 / √2 = √2⁄2
cos θ = Adjacent / Hypotenuse = 1 / √2 = √2⁄2

✅ Final Answer:
sin θ = √2⁄2
cos θ = √2⁄2

🔹Common Mistakes to Avoid

Confusing adjacent and opposite sides for the given angle.
Using degrees when radians are required, or vice versa.
Forgetting to simplify square roots.
Assuming the triangle is right-angled without confirmation.

🔹 Practice Questions

In a right triangle, if the opposite side is 7 cm and the adjacent side is 24 cm, find $\tan \theta$ .
Calculate $\sin 45^\circ$ , $\cos 60^\circ$ , and $\tan 30^\circ$ .
Find the hypotenuse if $\sin \theta = 0.6$ and the opposite side is 9 cm.
A ladder leans against a wall making a 60° angle with the ground. If the ladder is 10 m long, find the height reached on the wall.

MakemyIQ