Introduction to Trigonometry :

• Trigonometry is a branch of mathematics that measures distances and angles in triangles.
• It is useful in fields like Engineering, Astronomy, and Navigation.
• The term 'Trigonometry' comes from the Greek words 'Tri' meaning three, 'gona' meaning sides, and 'metron' meaning measurement.
• It involves determining the remaining angles and sides of a triangle by comparing the angles and sides given.
• The study of trigonometry is crucial in various fields like engineering, astronomy, and navigation.

Terms related to right angled triangle :

In right angled Δ ABC, ∠ B = 90°, ∠ A and ∠ C are acute angles.

Trigonometric ratios :

In Δ ABC, (below fig.)

(i) \(\frac{AC}{AB}=\frac{\text{Opposite side of ∠ B}}{Hypotenuse}\)

This ratio is called the 'sine' ratio of ∠ B and is written in brief as sin B.

(ii) \(\frac{BC}{AB}=\frac{\text{Adjacent side of ∠ B}}{Hypotenuse}\)

This ratio is called the 'cosine' ratio of ∠ B and is written in brief as cos B.

(iii) \(\frac{AC}{BC}=\frac{\text{Opposite side of ∠ B}}{\text{Adjacent side of ∠ B}}\)

This ratio is called the 'tangent' ratio of ∠ B and is written in brief as tan B.

Sometimes we write measures of acute angles of a right angled triangle by using Greek letters θ (Theta), α (Alpha), β (Beta) etc.

In the below figure of Δ ABC, measure of acute angle C is denoted by the letter θ.

So we can write the ratios sin C, cos C, tan C as sin θ, cos θ, tan θ respectively.

sin C = sin θ = \(\frac{AB}{AC}\),

cos C = cos θ = \(\frac{BC}{AC}\),

tan C = tan θ = \(\frac{AB}{BC}\),

Remember :

sin ratio = \(\frac{\text{Opposite side}}{Hypotenuse}\)       sin θ = \(\frac{\text{Opposite side of ∠ θ}}{Hypotenuse}\)

cos ratio = \(\frac{\text{Adjacent side}}{Hypotenuse}\)       cos θ = \(\frac{\text{Adjacent side of ∠ θ}}{Hypotenuse}\)

tan ratio = \(\frac{\text{Opposite side}}{\text{Adjacent side}}\)        tan θ = \(\frac{\text{Opposite side of ∠ θ}}{\text{Adjacent side of ∠ θ}}\)

Relation among trigonometric ratios :

cos (90 - θ) = sin θ,

sin (90 - θ) = cos θ

\(\frac{ sin\,θ}{cos\,θ}\) = tan θ,

tan θ × tan (90 - θ) = 1

cosec θ, sec θ and cot θ are inverse ratios of sin θ, cos θ and tan θ respectively.

cosec θ = \(\frac{1}{sin\,θ}\), sec θ = \(\frac{1}{cos\,θ}\), cot θ = \(\frac{ cos\,θ}{sin\,θ}\)

sec θ = cosec (90 - θ), cosec θ = sec (90 - θ),

tan θ = cot (90 - θ),  cot θ = tan (90 - θ)

Trigonometric ratios of 30° and 60° angles :

Theorem of 30°- 60°-90° triangle :

We know that if the measures of angles of a triangle are 30°,60°, 90° then side opposite to 30° angle is half of the hypotenuse and side opposite to 60° angle is \(\frac{\sqrt{3}}{2}\) of hypotenuse.

In the Fig. Δ ABC is a right angled triangle.

∠ C = 30°, ∠ A = 60°, ∠ B = 90°.

∴ AB = \(\frac{1}{2}\) AC and BC = \(\frac{\sqrt{3}}{2}\) AC

 

Measures of angles → ……………. Ratios ↓	0°	30°	45°	60°	90°
sin	0	\(\frac{1}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{\sqrt{3}}{2}\)	1
cos	1	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{1}{2}\)	0
tan	0	\(\frac{1}{\sqrt{3}}\)	1	\(\sqrt{3}\)	Undefined

 

Prove : (sin θ)² + (cos θ)² = 1.

Proof :

Δ PQR is a right angled triangle.

∠ PQR = 90°, ∠ R = θ

sin θ = \(\frac{PQ}{PR}\)           ..........(I)

and cos θ = \(\frac{QR}{PR}\) ..........(II)

Using Pythagoras’ theorem,

PQ² + QR² = PR²

∴  \(\frac{(PQ)^2}{(PR)^2}+\frac{(QR)^2}{(PR)^2}=\frac{(PR)^2}{(PR)^2}\) .... dividing each term by PR²

∴ \(\frac{(PQ)^2}{(PR)^2}+\frac{(QR)^2}{(PR)^2}\) = 1

∴ (sin θ)² + (cos θ)² = 1    …..from (I) & (II)

Important Equation in Trigonometry :

(sin θ)² + (cos θ)² = 1.

(sin θ)² means square of sin θ. It is written as sin² θ.

∴ sin² θ + cos² θ = 1.

This equation is true even when θ = 0° or θ = 90°.

(i) 0≤ sin θ ≤ 1,0 ≤ sin² θ ≤ 1

(ii) 0≤ cos θ ≤ 1,0 ≤ cos² θ ≤ 1.