Cos 15pi/4
The value of cos 15pi/4 is 0.7071067. . .. Cos 15pi/4 radians in degrees is written as cos ((15π/4) × 180°/π), i.e., cos (675°). In this article, we will discuss the methods to find the value of cos 15pi/4 with examples.
- Cos 15pi/4: 1/√2
- Cos 15pi/4 in decimal: 0.7071067. . .
- Cos (-15pi/4): 0.7071067. . . or 1/√2
- Cos 15pi/4 in degrees: cos (675°)
What is the Value of Cos 15pi/4?
The value of cos 15pi/4 in decimal is 0.707106781. . .. Cos 15pi/4 can also be expressed using the equivalent of the given angle (15pi/4) in degrees (675°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 15pi/4 radians = 15pi/4 × (180°/pi) = 675° or 675 degrees
∴ cos 15pi/4 = cos 15π/4 = cos(675°) = 1/√2 or 0.7071067. . .
Explanation:
For cos 15pi/4, the angle 15pi/4 > 2pi. Given the periodic property of the cosine function, we can represent it as cos(15pi/4 mod 2pi) = cos(7pi/4). The angle 15pi/4, coterminal to angle 7pi/4, is located in the Fourth Quadrant(Quadrant IV).
Since cos function is positive in the 4th quadrant, thus cos 15pi/4 value = 1/√2 or 0.7071067. . .
Similarly, cos 15pi/4 can also be written as, cos 15pi/4 = (15pi/4 + n × 2pi), n ∈ Z.
⇒ cos 15pi/4 = cos 23pi/4 = cos 31pi/4, and so on.
Note: Since, cosine is an even function, the value of cos(-15pi/4) = cos(15pi/4).
Methods to Find Value of Cos 15pi/4
The cosine function is positive in the 4th quadrant. The value of cos 15pi/4 is given as 0.70710. . .. We can find the value of cos 15pi/4 by:
- Using Trigonometric Functions
- Using Unit Circle
Cos 15pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 15pi/4 as:
- ± √(1-sin²(15pi/4))
- ± 1/√(1 + tan²(15pi/4))
- ± cot(15pi/4)/√(1 + cot²(15pi/4))
- ±√(cosec²(15pi/4) - 1)/cosec(15pi/4)
- 1/sec(15pi/4)
Note: Since 15pi/4 lies in the 4th Quadrant, the final value of cos 15pi/4 will be positive.
We can use trigonometric identities to represent cos 15pi/4 as,
- -cos(pi - 15pi/4) = -cos(-11pi/4)
- -cos(pi + 15pi/4) = -cos 19pi/4
- sin(pi/2 + 15pi/4) = sin 17pi/4
- sin(pi/2 - 15pi/4) = sin(-13pi/4)
Cos 15pi/4 Using Unit Circle
To find the value of cos 15π/4 using the unit circle, represent 15pi/4 in the form (1 × 2pi) + 7pi/4 [∵ 15pi/4>2pi] ∵ cosine is a periodic function, cos 15pi/4 = cos 7pi/4.
- Rotate ‘r’ anticlockwise to form 7pi/4 or 15pi/4 angle with the positive x-axis.
- The cos of 15pi/4 equals the x-coordinate(0.7071) of the point of intersection (0.7071, -0.7071) of unit circle and r.
Hence the value of cos 15pi/4 = x = 0.7071 (approx)
☛ Also Check:
FAQs on Cos 15pi/4
What is Cos 15pi/4?
Cos 15pi/4 is the value of cosine trigonometric function for an angle equal to 15π/4 radians. The value of cos 15pi/4 is 1/√2 or 0.7071 (approx)
What is the Value of Cos 15pi/4 in Terms of Cot 15pi/4?
We can represent the cosine function in terms of the cotangent function using trig identities, cos 15pi/4 can be written as -cot(15pi/4)/√(1 + cot²(15pi/4)). Here, the value of cot 15pi/4 is equal to -1.
How to Find Cos 15pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 15pi/4 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(15pi/4))
- ± 1/√(1 + tan²(15pi/4))
- ± cot(15pi/4)/√(1 + cot²(15pi/4))
- ±√(cosec²(15pi/4) - 1)/cosec(15pi/4)
- 1/sec(15pi/4)
☛ Also check: trigonometry table
What is the Value of Cos 15pi/4 in Terms of Sec 15pi/4?
Since the secant function is the reciprocal of the cosine function, we can write cos 15pi/4 as 1/sec(15pi/4). The value of sec 15pi/4 is equal to 1.414213.
How to Find the Value of Cos 15pi/4?
The value of cos 15pi/4 can be calculated by constructing an angle of 15π/4 radians with the x-axis, and then finding the coordinates of the corresponding point (0.7071, -0.7071) on the unit circle. The value of cos 15pi/4 is equal to the x-coordinate (0.7071). ∴ cos 15pi/4 = 0.7071.