Sin 45 Degrees

The value of sin 45 degrees is 0.7071067. . .. Sin 45 degrees in radians is written as sin (45° × π/180°), i.e., sin (π/4) or sin (0.785398. . .). In this article, we will discuss the methods to find the value of sin 45 degrees with examples.

• Sin 45°: 0.7071067. . .
• Sin 45° in fraction: 1/√2
• Sin (-45 degrees): -0.7071067. . .
• Sin 45° in radians: sin (π/4) or sin (0.7853981 . . .)

What is the Value of Sin 45 Degrees?

The value of sin 45 degrees in decimal is 0.707106781. . .. Sin 45 degrees can also be expressed using the equivalent of the given angle (45 degrees) in radians (0.78539 . . .).

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 45 degrees = 45° × (π/180°) rad = π/4 or 0.7853 . . .
∴ sin 45° = sin(0.7853) = 1/√2 or 0.7071067. . .

Explanation:

For sin 45 degrees, the angle 45° lies between 0° and 90° (First Quadrant). Since sine function is positive in the first quadrant, thus sin 45° value = 1/√2 or 0.7071067. . .
Since the sine function is a periodic function, we can represent sin 45° as, sin 45 degrees = sin(45° + n × 360°), n ∈ Z.
⇒ sin 45° = sin 405° = sin 765°, and so on.
Note: Since, sine is an odd function, the value of sin(-45°) = -sin(45°).

Methods to Find Value of Sin 45 Degrees

The sine function is positive in the 1st quadrant. The value of sin 45° is given as 0.70710. . .. We can find the value of sin 45 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

Sin 45 Degrees Using Unit Circle

To find the value of sin 45 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form a 45° angle with the positive x-axis.
• The sin of 45 degrees equals the y-coordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.

Hence the value of sin 45° = y = 0.7071 (approx)

Sin 45° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin 45 degrees as:

• ± √(1-cos²(45°))
• ± tan 45°/√(1 + tan²(45°))
• ± 1/√(1 + cot²(45°))
• ± √(sec²(45°) - 1)/sec 45°
• 1/cosec 45°

Note: Since 45° lies in the 1st Quadrant, the final value of sin 45° will be positive.

We can use trigonometric identities to represent sin 45° as,

• sin(180° - 45°) = sin 135°
• -sin(180° + 45°) = -sin 225°
• cos(90° - 45°) = cos 45°
• -cos(90° + 45°) = -cos 135°

☛ Also Check:

• sin 30 degrees
• sin 104 degrees
• sin 360 degrees
• sin 255 degrees
• sin 910 degrees
• sin 220 degrees

FAQs on Sin 45 Degrees

What is Sin 45 Degrees?

Sin 45 degrees is the value of sine trigonometric function for an angle equal to 45 degrees. The value of sin 45° is 1/√2 or 0.7071 (approx).

How to Find the Value of Sin 45 Degrees?

The value of sin 45 degrees can be calculated by constructing an angle of 45° with the x-axis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of sin 45° is equal to the y-coordinate (0.7071). ∴ sin 45° = 0.7071.

How to Find Sin 45° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin 45° can be given in terms of other trigonometric functions as:

• ± √(1-cos²(45°))
• ± tan 45°/√(1 + tan²(45°))
• ± 1/√(1 + cot²(45°))
• ± √(sec²(45°) - 1)/sec 45°
• 1/cosec 45°

☛ Also check: trigonometric table

What is the Exact Value of sin 45 Degrees?

The exact value of sin 45 degrees can be given accurately up to 8 decimal places as 0.70710678 and 1/√2 in fraction.

What is the Value of Sin 45 Degrees in Terms of Cot 45°?

We can represent the sine function in terms of the cotangent function using trig identities, sin 45° can be written as 1/√(1 + cot²(45°)). Here, the value of cot 45° is equal to 1.