Cos Theta Formula

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Amongst all the trigonometric formulas, the most important ones are the right triangle formulas. The Cos θ = Adjacent / Hypotenuse

Cos angle formula

There are many formulas in trigonometry but there are few most important basic formulas in trigonometry when it comes to a right-angle triangle. The Cos theta or cos θ is the ratio of the adjacent side to the hypotenuse, where θ is one of the acute angles. The cosine formula is as follows:

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.

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.

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

Cos Theta Formula Questions

Example 1:

If Sin x = 4/5, Find the value of Cos x?

Solution: Using Trigonometric identities: Cos2x = 1- Sin2x

Cos2x = 1 – (4/5)2

= 1 – 16/25

= (25 – 16) / 25

= 9/25

Cos x = √9/259/25

= 3/5

Example 2: If Sec x = 4/7, find the Cos x?

Solution: As cos x = 1/sec x

Therefore, cos x = 1/ 4/7

= 7/4

Cos Square Theta Formula

Trigonometric identities are equations that relate to different trigonometric functions and are true for any value of the variable that is there in the domain. Basically, an identity is an equation that holds true for all the values of the variables present in it. The function of an angle i.e the angles and sides relationships is given by trigonometric functions. Sine, cosine, tangent, cotangent, Cos, Cosec are called the trigonometric functions. Let’s look into the Cos Square theta formula below.

Formula for Cos Square theta

According to the trigonometric identities, we know that,

cos2θ + sin2θ = 1

where, 

  • θ is an acute angle of a right triangle.
  • sinθ and cosθ are the trigonometric ratios given as follows:
    sinθ = Altitude/Hypotenuse
    cosθ = Base/Hypotenuse
  • sin2θ is the square of sinθ and cos2θ is the square of cosθ i.e,
    sin2θ = (sinθ)2
    cos2θ= (cosθ)2

Thus cos square theta formula is given by,

cos2θ = 1 – sin2θ

Solved Examples using Cos Square Theta Formula

Example 1: What is the value of cos square x, if Sin x = 4/5 ?

Solution:

Using Cos Square theta formula,

Cos 2 x = 1 – Sin2 x

= 1 – (4/5)2

= 1 – 16/25

= (25 – 16) / 25

= 9/25

Thus, cos x = 3/5

Example 2: If cos2x – sin2x = 41/841, then find the value of cos2x.

Solution:

Given: cos2x – sin2x = 41/841

We know that,

sin2x = 1 – cos2

Substituting in the above equation we get,

cos2x – (1 – cos2x) = 41/841

⇒2 cos2x – 1 = 41/841

⇒2 cos2x = 1 + 41/841

⇒2 cos2x = 882/841

⇒cos2x = 882/(841 × 2)

⇒cos2x = 441/841

Thus, the value of cos2x is 441/841.

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