Cos Theta Formula

Introduction to Cos Theta Formula

In Mathematics, there are a total of six different types of trigonometric functions which are sine (sin), Cosine (cos), Secant (sec), Cosecant (cosec), Tangent (tan), and Cotangent (cot). All these six different types of trigonometric functions symbolize the relationship between the ratios of different sides of a right-angle triangle. These trigonometric functions are important for studying triangles, height, and distance, light, sound, wave, etc. The theta formula for different trigonometric functions is different, Theta is represented by θ.


In a Right-Angled Triangle

  • Sine (θ) = Opposite/Hypotenuse

  • Cos (θ) = Adjacent/Hypotenuse

  • Tan (θ) = Opposite/Adjacent

  • Cot (θ) = Adjacent/Opposite

  • Cosec (θ) = Hypotenuse/Opposite

  • Sec (θ) = Hypotenuse/Adjacent

In this topic, we will discuss what is cos theta and the values of different angles.


Cos Angle Formula

In a right-angled triangle. The Cos theta or cos θ is the ratio of the adjacent side to the hypotenuse.


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In the given right angle triangle A is an adjacent side, O is perpendicular and H represents the hypotenuse.

Cos θ = Adjacent/Hypotenuse

Here θ represents the angle of a triangle. The angles by which trigonometric functions can be represented are called trigonometry angles. The important angles of trigonometry are 0°, 30°, 45°, 60°, 90°. All of these are standard angles of trigonometric ratios, such as sin, cos, tan, sec, cosec, and cot. Each of these angles has different values with different trigonometric functions. 


θ

30°

45°

60°

90°

sin θ

0

1/2

1/√2

√3/2

1

cos θ

1

√3/2

1/√2

1/2

0

tan θ

0

1/√3

1

√3

Undefined

cosec θ

Undefined

2

√2

2/√3

1

sec θ

1

2/√3

√2

2

Undefined

cot θ

Undefined

√3

1

1/√3

0


Note: 1 cos theta = 1. Cos θ;  Eg: 1 cos 30° = 1. Cos 30° = 1 x  √3/2 = √3/2

And cos θ = 1/sec θ

Or, sec θ = 1/cos θ

Also, sin (90 -  θ) = cos θ and Cos (90 -  θ) = sin θ.

Also remember sin 45 = cos 45 = 1/√2. The value of sin θ and cos θ can never be greater than 1.

FAQs (Frequently Asked Questions)

Q1:  If Sin θ = 3/5, What will be the Value of Cos Theta?

Answer: Using Trigonometric identities: Cos2θ + Sin2θ = 1,

so Cos2θ = 1- Sin2θ

Cos2θ = 1 – (3/5)2

1 – (9/25)

= (25 – 9)/25

= 16/25

Cos θ = √(16/25)

Cos θ = 4/5

So, cos theta is equal to 4/5.

Q2: If sin 3x = cos (x - 26°), Where 3x is an Acute Angle, Find the Value of A.

Answer: Given that, sin 3x = cos (x - 26°) ….(1)

Since, sin 3x = cos (90° – 3x), we can write (1) as:

cos(90°- 3x) = cos (x - 26°)

Since, 90°-3x = x – 26°

Therefore,

90° + 26° = 3x + x

4 x = 116°

x = 116° / 4 = 29°

Therefore, the value of x is 29°.

Q3: If cos x = 4/7, Find the Sec x?

Answer: Since we know that cos x = 1/sec x

And sec x = 1/cos x

Therefore, sec x = 1/4/7

= 7/4