Question

$\sf \red {What \: is \: Trigonometry ?}$
$\sf \blue {Give \: 2 - 3 \: example}$
$\sf \green {Example \: should \: must \: contain \: full \: explanation}$
$\sf Must \: and \: should \: Explain \: clearly .$
​

Answer 1

Solution :-

Trigonometry is nothing just a way to find out the measurement of sides and angles of a triangle.

• The first concept of trigonometry deals with trigonometric ratios

Example 1 :-

In ΔABC, right angled at A, if AB = 5 , AC = 12 and Hypotenuse = BC = 13 find sinB and tanB

Solution :-

Base AB = 5 , Perpendicular AC = 12 and

Hypotenuse BC = 13

Sin B = AC / BC = 12/13

Tan B = AC/ AB = 12/5

• The second concept of trigonometry deals with trigonometric ratios of complementary angles

-Sin( 90° - A) = CosA

-Cos( 90° - A) = sinA

-tan( 90° - A) = cotA

-Cot( 90° - A) = tanA

- sec( 90° - A) = CosecA

- Cosec( 90° - A) = secA

Example 2 :-

Evaluate the following

Sin39° - cos51°

Solution :-

We have, Sin39° - Cos51°

= Sin( 90° - 51°) - cos51°

= Cos51° - Cos51°

= 0

• The third concept of trignometry deals with trigonometric identities

Example 3 :-

Prove that,

( 1 - sin²Φ)sec²Φ = 1

Solution :-

(1 - sin²Φ)sec²Φ = 1

By taking LHS,

( 1 - sin²Φ )sec²Φ

[ By using identity Sin²Φ + Cos²Φ = 1 ]

= cos²Φ × sec²Φ

= cos²Φ × 1/cos²Φ

= 1

Answer 2

Answer:-

Trigonometry:-

• The word trigonometry can be splitted into 3 words ⟶ Tri , gonia , metron

1. Tri = Three
2. Gonia = Angle
3. Metron = measurement.

• Thus, trigonometry can be defined as the measurement of three angles of a triangle.

• It was derived from Greek.

Trigonometric Ratios:-

Let us take a right angled triangle ABC , where ∠B = 90° and let an angle θ be at C.

From the figure we can say that, AB is the opposite side of θ , BC is the adjacent side and the longest side is the Hypotenuse.

Now,

• sine of θ = sin θ = AB/AC = Opposite side/ Hypotenuse

• cosine of θ = cos θ = BC/AC = Adjacent side/Hypotenuse

• tangent of θ = tan θ = AB/BC = Opposite side/Adjacent side

• cotangent of θ = cot θ = BC/AB = Adjacent side/Opposite side

• secant of θ = sec θ = AC/BC = Hypotenuse/Adjacent side

• cosecant of θ = cosec θ = AC/AB = Hypotenuse/Opposite side.

We use these 6 ratios for finding an angle or the ratio of any two sides of the given triangle.

Also, from the above relations we can say that:

1. cosec θ = 1/sin θ
2. sec θ = 1/cos θ
3. cot θ = 1/tan θ.

Note:-

We use Pythagoras Theorem for finding any side of a right angled triangle if it's other two sides are given.

⟹ (Hypotenuse)² = (Opposite side)² + (Adjacent side)²

Examples:-

1) In a right angled triangle PQR , right angled at Q ; ∠R = θ , PQ = 4 m , QR = 3 m and PR = 5 m. Then find the value of cos θ.

Answer:-

We are given:

• ∠Q = 90°
• ∠R = θ
• PQ = 4 m
• QR = 3 m
• PR = 5 m.

We know that;

Cos θ = Adjacent side/Hypotenuse

Here, Adjacent side = QR and Hypotenuse = PR.

So, cos θ = QR/PR.

substitute the given values.

⟹ cos θ = 3/5

∴ The value of cos θ is 3/5.

________________________________

2) From the above question , find sin θ.

Answer:-

We know that;

sin θ = Opposite side/Hypotenuse.

Here, Opposite side = PQ and Hypotenuse= PR.

so,

⟹ sin θ = PQ/PR

⟹ sin θ = 4/5

Hence, the value of sin θ is 4/5.

_________________________________

3) If sin θ = 1/2 then find cosec θ.

Answer:-

We have:

• sin θ = 1/2.

We know that,

cosec θ = 1/sin θ

So,

⟹ cosec θ = 1 / (1/2)

⟹ cosec θ = 2

∴ The value of cosec θ is 2.

(Refer the attachment for the diagrams.)