# Cos 37 Degrees

Created by: Team Maths - Examples.com, Last Updated: May 29, 2024

## Cos 37 Degrees

he value of cos 37 degrees is approximately 0.7986355. In radians, cos 37 degrees is written as cos (37° × π/180°), which is cos (0.6457718…). This article will discuss various methods to find the value of cos 37 degrees, with examples.

• Cos 37°: 0.7986355.
• Cos (-37°): 0.7986355, due to the even property of the cosine function.
• Cos 37° in Radians (37° × π/180°): cos (0.6457718…).

## What is the Value of Cos 37 Degrees?

The cosine of 37 degrees is approximately:

• Cos 37° = 0.7986355

In radians, 37 degrees is converted using the formula: degrees × π/180.

• 37 degrees in radians = 37° × π/180 ≈ 0.6457718

• Cos 37° = cos (0.6457718)

## Methods to Find Value of Cos 37 Degrees

1. Using Trigonometric Functions
2. Using the Unit Circle

### Using Trigonometric Identities

#### 1. Pythagorean Identity

The Pythagorean identity relates the cosine and sine of an angle.

• Formula: Cos 37° = √(1 – sin² 37°)

#### Example:

If sin 37° is approximately 0.6018,

• Cos 37° = √(1 – 0.6018²) ≈ 0.7986

#### 2. Tangent Function

The tangent function can also help in finding the cosine of an angle.

• Formula: Cos 37° = 1 / √(1 + tan² 37°)

#### Example:

If tan 37° is approximately 0.7536,

• Cos 37° = 1 / √(1 + 0.7536²) ≈ 0.7986

#### 3. Cotangent Function

Cotangent, another trigonometric function, can be used as follows:

• Formula: Cos 37° = cot 37° / √(1 + cot² 37°)

#### Example:

If cot 37° is approximately 1.327,

• Cos 37° = 1.327 / √(1 + 1.327²) ≈ 0.7986

#### 4. Cosecant Function

Cosecant provides another way to express cos 37 degrees:

• Formula: Cos 37° = √(cosec² 37° – 1) / cosec 37°

#### Example:

If cosec 37° is approximately 1.6629,

• Cos 37° = √(1.6629² – 1) / 1.6629 ≈ 0.7986

#### 5. Secant Function

Finally, secant, the reciprocal of cosine, gives us a direct formula:

• Formula: Cos 37° = 1 / sec 37°

#### Example:

If sec 37° is approximately 1.251,

• Cos 37° = 1 / 1.251 ≈ 0.7986

### Angle Relationships

Cos 37 degrees can be related to other angles using complementary and supplementary angle identities:

1. Supplementary Angles:
• Cos 37° is equal to negative cos (180° minus 37°), which is negative cos 143°.
• Formula: cos 37° = -cos (180° – 37°) = -cos 143°
2. Angles Beyond 180 Degrees:
• Cos 37° is equal to negative cos (180° plus 37°), which is negative cos 217°.
• Formula: cos 37° = -cos (180° + 37°) = -cos 217°
3. Complementary Angles:
• Cos 37° is equal to the sine of 90° minus 37°, which is sine 53°.
• Formula: cos 37° = sin (90° – 37°) = sin 53°
4. Angles Plus 90 Degrees:
• Cos 37° is equal to the sine of 90° plus 37°, which is sine 127°.
• Formula: cos 37° = sin (90° + 37°) = sin 127°

### Practical Example

#### Example: Using the Pythagorean Identity

Given that sin 37° is approximately 0.6018, we can find cos 37°:

• Cos 37° is equal to the square root of 1 minus the sine squared of 37°.
• Cos 37° = √(1 – sin² 37°) = √(1 – 0.6018²) ≈ 0.7986

## Cos 37 Degrees Using Unit Circle

### Step 1: Understand the Unit Circle

• The unit circle has a radius of 1 and is centered at the origin (0, 0).
• It is divided by the x-axis and y-axis, which intersect the circle at four key points: (1, 0), (-1, 0), (0, 1), and (0, -1).

### Step 2: Locate the Angle

• Identify the 37-degree angle on the unit circle. This angle starts from the positive x-axis and rotates counterclockwise.

### Step 3: Find the Intersection Point

• Draw the terminal side of the 37-degree angle. This line intersects the unit circle.
• The coordinates of this intersection point represent the cosine and sine of 37 degrees.
• According to the image, the intersection point is approximately (0.7986, 0.6018).

### Step 4: Determine the Cosine Value

• The x-coordinate of the intersection point is the value of cos 37°.
• From the image, cos 37° ≈ 0.7986.

### Step 5: Confirm Positivity in the 1st Quadrant

• Since 37 degrees lies in the 1st quadrant, both cosine and sine values are positive.

## Solved Problems

Problem 1: Verify Cos 37° using the Pythagorean identity

Solution:
We know that:
Sin 37° = 3/5
Using the Pythagorean identity:
Cos² 37° + Sin² 37° = 1
Substitute Sin 37°:
Cos² 37° + (3/5)² = 1
Cos² 37° + 9/25 = 1
Cos² 37° = 1 – 9/25
Cos² 37° = 25/25 – 9/25
Cos² 37° = 16/25
Cos 37° = 4/5

Problem 2: Use Cos 37° to find Sin 53°

Solution:
We know that:
Sin 53° = Cos (90° – 53°) = Cos 37°
Cos 37° = 4/5
Therefore:
Sin 53° = 4/5

Problem 3: Calculate the value of 2Cos² 37° – 1

Solution:
First, calculate Cos² 37°:
Cos 37° = 4/5
Cos² 37° = (4/5)² = 16/25
Then, use the expression:
2Cos² 37° – 1 = 2 × 16/25 – 1
2Cos² 37° – 1 = 32/25 – 1
2Cos² 37° – 1 = 32/25 – 25/25
2Cos² 37° – 1 = 7/25

Problem 4: Find Cos 37° given Tan 37°

Solution:
We know:
Tan 37° = 3/4
Using the identity:
Tan x = Sin x / Cos x
Tan 37° = (3/5) / Cos 37°
Cos 37° = 4/5

Problem 5: Use Cos 37° to find Cos 74°

Solution:
We know that:
Cos 74° = Cos (2 × 37°)
Using the double-angle identity:
Cos 2x = 2Cos² x – 1
Cos 74° = 2Cos² 37° – 1
Cos² 37° = 16/25
So:
Cos 74° = 2 × 16/25 – 1
Cos 74° = 32/25 – 1
Cos 74° = 32/25 – 25/25
Cos 74° = 7/25
Cos 74° = 7/25

## What is the value of cos 37 degrees in fractions?

The value of cos 37 degrees is approximately 4/5, but more accurately, it is 0.7986, which is close to the fraction 4/5.

## What is the value of sin 37 degrees?

The value of sin 37 degrees is approximately 0.6018, often memorized in trigonometry for quick reference.

## What is the magnitude of cos 37?

The magnitude of cos 37 degrees is 0.7986, representing its absolute value, which is always positive.

## What is sin 37 in cosine?

Sin 37 degrees can be expressed using cosine as sin 37° = cos 53°, based on complementary angles.

## How to remember sin 37?

Remember sin 37 degrees as approximately 0.6, part of the commonly used sin-cosine pair: sin 37° ≈ 0.6 and cos 37° ≈ 0.8.

Text prompt