Question

Convert 1470 from degrees to radians. Then find the coterminal angle what would be between 0 and 2T radians. Finally give the exact cos of this angle. Do not use decimals in your answers.

You answer will have 3 parts, again do not use decimals:
-The original angle converted to radians, showing the steps used.
-The coterminal angle that is between 0 and 2 radians, showing the steps used.
-The exact cos of this angle.

Answer 1

Let's go through each part step by step:

1. Converting 1470 degrees to radians:

To convert degrees to radians, we use the formula: Radians = Degrees × π / 180

Given: Degrees = 1470

Radians = 1470 × π / 180

Calculating the value:

Radians = 1470 × 3.14159 / 180

Radians = 25.6535898

Therefore, the original angle of 1470 degrees is equivalent to 25.6535898 radians.

2. Finding the coterminal angle between 0 and 2π radians:

To find the coterminal angle, we can subtract or add multiples of 2π until we get an angle between 0 and 2π.

Given: Radians = 25.6535898

Subtracting multiples of 2π:

25.6535898 - (2π) = 25.6535898 - (2 × 3.14159) = 25.6535898 - 6.28318 = 19.3704098

Therefore, the coterminal angle between 0 and 2π radians is 19.3704098 radians.

3. Finding the exact cosine of the coterminal angle:

To find the exact cosine of the coterminal angle, we use the unit circle. The cosine value represents the x-coordinate of the point on the unit circle.

Given: Coterminal Angle = 19.3704098 radians

Using the unit circle:

Since the angle is positive and between 0 and 2π, we can determine the cosine by looking at the x-coordinate of the corresponding point on the unit circle.

The exact cosine of 19.3704098 radians is cos(19.3704098) = cos(2π - 19.3704098) = cos(2.4711858) = -0.7933533403

Therefore, the exact cosine of the coterminal angle is -0.7933533403.

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Answer 2

To convert 1470 degrees to radians, we use the conversion factor that 180 degrees is equal to π radians.

1) Converting 1470 degrees to radians:
1470 degrees * (π radians / 180 degrees) = 1470π/180 radians

Therefore, the original angle of 1470 degrees is equal to (49π/6) radians.

2) Finding the coterminal angle between 0 and 2π radians:
To find the coterminal angle between 0 and 2π radians, we need to subtract or add multiples of 2π to the original angle.

(49π/6) radians + 2π = (49π/6) + (12π/6) = (61π/6) radians

Therefore, the coterminal angle between 0 and 2π radians is (61π/6) radians.

3) Finding the exact cosine of the coterminal angle:
The cosine of an angle can be determined using the unit circle or trigonometric identities. Since the angle is given in terms of π, we can use the cosine values of common angles in the unit circle.

The exact cosine of (61π/6) radians can be written as:
cos(61π/6) = cos((10π + π/6))

In the unit circle, cos(π/6) = √3/2

Therefore, the exact cosine of (61π/6) radians is:
cos(61π/6) = cos(10π + π/6) = cos(π/6) = √3/2

So, the exact cosine of the coterminal angle is √3/2.