Trigonometric Ratios and Identities

 📌 What Are Trigonometric Ratios?

Trigonometric Ratios relate the angles of a right-angled triangle to the ratios of its sides. These are fundamental in solving geometry-based quantitative problems.

Let θ be an acute angle in a right triangle:

• Hypotenuse (H): The side opposite the right angle

• Opposite Side (O): Side opposite to angle θ

• Adjacent Side (A): Side adjacent to angle θ (other than hypotenuse)

🔻 Basic Trigonometric Ratios:

Ratio	Formula	Also written as
sin θ	Opposite / Hypotenuse	O/H
cos θ	Adjacent / Hypotenuse	A/H
tan θ	Opposite / Adjacent	O/A
cot θ	Adjacent / Opposite	A/O = 1/tan θ
sec θ	Hypotenuse / Adjacent	H/A = 1/cos θ
cosec θ	Hypotenuse / Opposite	H/O = 1/sin θ

📌 Standard Values of Trigonometric Ratios

θ (degrees)	sin θ	cos θ	tan θ
0°	0	1	0
30°	½	√3⁄2	1⁄√3
45°	1⁄√2	1⁄√2	1
60°	√3⁄2	½	√3
90°	1	0	∞

Tip: Remember the mnemonic → sin: 0, ½, 1/√2, √3/2, 1

(then reverse for cos)

📌 Trigonometric Identities

These are algebraic relations that are always true for any value of θ.

🔹 Fundamental Identities:

1. sin²θ + cos²θ = 1

2. 1 + tan²θ = sec²θ

3. 1 + cot²θ = cosec²θ

🔹 Reciprocal Identities:

• sin θ = 1 / cosec θ

• cos θ = 1 / sec θ

• tan θ = 1 / cot θ (and vice versa)

🔹 Quotient Identities:

• tan θ = sin θ / cos θ

• cot θ = cos θ / sin θ

📌 Co-Function Identities

• sin(90° – θ) = cos θ

• cos(90° – θ) = sin θ

• tan(90° – θ) = cot θ

• cot(90° – θ) = tan θ

• sec(90° – θ) = cosec θ

• cosec(90° – θ) = sec θ

📝 These are useful in angle transformation questions.

📌 Negative Angle Identities

• sin(–θ) = –sin θ

• cos(–θ) = cos θ

• tan(–θ) = –tan θ

• cot(–θ) = –cot θ

• sec(–θ) = sec θ

• cosec(–θ) = –cosec θ

Note: Sine, tangent, cotangent, and cosecant are odd functions; cosine and secant are even.

📌 Sum, Difference, Product Trigonometric Ratios Identities

🔷 Sum and Difference Formulas

✅ Sine Formulas

                          sin (A + B) = sin A cos B + cos A sin B

                          sin (A - B) = sin A cos B - cos A sin B

✅ Cosine Formulas        

                          cos (A + B) = cos A cos B - sin A sin B
                          cos (A - B) = cos A cos B + sin A sin B
✅ Tangent Formulas          

                          tan (A + B) = (tan A + tan B)/ (1 - tan A tan B)
                          tan (A - B) = (tan A - tan B)/ (1 + tan A tan B)
✅ Cotangent Formulas

                          cot (A + B) = (cot A cot B - 1)/(cot B - cot A)
                          cot (A - B) = (cot A cot B + 1)/(cot B - cot A)

🔷Product-to-Sum Identities

                          2 sin A⋅cos B = sin(A + B) + sin(A - B)

                          2 cos A⋅cos B = cos(A + B) + cos(A - B)

                          2 sin A⋅sin B = cos(A - B) - cos(A + B)

📌 Complementary Angle Shortcuts

These can quickly help in solving problems without using calculators:

• sin 30° = cos 60° = ½

• sin 45° = cos 45° = 1⁄√2

• sin 60° = cos 30° = √3⁄2

📌 Examples

Example 1:
If sin θ = 3⁄5, find cos θ and tan θ.

Solution:
sin θ = Opposite / Hypotenuse = 3/5
⇒ Opposite = 3, Hypotenuse = 5
Using Pythagoras: Adjacent = √(5² – 3²) = √(25 – 9) = √16 = 4

⇒ cos θ = 4/5
⇒ tan θ = 3/4

Example 2:
Prove that 1 + tan² θ = sec² θ

Solution:
From definitions:
tan θ = sin θ / cos θ
sec θ = 1 / cos θ

So,
1 + tan² θ = 1 + (sin² θ / cos² θ)
= (cos² θ + sin² θ) / cos² θ
= 1 / cos² θ = sec² θ ✅

📌 Tips and Tricks

✅ Memorize values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90°
✅ Use identities to convert and simplify complex expressions
✅ In exams, always reduce the problem using identities wherever possible
✅ SOH-CAH-TOA helps remember the basic ratios