Cos 40 degrees

Cos 40 degrees DEFAULT

cos(40)

Enter angle in degrees or radians:


Calculate cos(40)°

Determine quadrant:

Since our angle is between 0 and 90 degrees, it is located in Quadrant I
In the first quadrant, the values for sin, cos and tan are positive.

Determine angle type:

40 is an acute angle since it is less than 90°
cos(40) = 0.76604444363175

Write cos(40) in terms of sin

Since 40° is less than 90, we can express this in terms of a cofunction
cos(θ) = sin(90 - θ)
cos(40) = sin(90 - 40)
cos(40) = sin(50)


In Microsoft Excel or Google Sheets, you write this function as =COS(RADIANS(40))

Trigonometric Function Values of Special Angles

θ°θradianssin(θ)cos(θ)tan(θ)csc(θ)sec(θ)cot(θ)
0010010
30°π/61/2√3/2√3/322√3/3√3
45°π/4√2/2√2/21√2√21
60°π/3√3/21/2√32√3/32√3/3
90°π/210N/A10N/A
120°2π/3√3/2-1/2-√32√3/3-2-√3/3
135°3π/4√2/2-√2/2-1√2-√2-1
150°5π/61/2-√3/2-√3/32-2√3/3-√3
180°π0-100-1N/A
210°7π/6-1/2-√3/2√3/3-2-2√3/3√3
225°5π/4-√2/2-√2/21-√2-√21
240°4π/3-√3/2-1/2√3-2√3/3-2√3/3
270°3π/2-10N/A-10N/A
300°5π/3-√3/21/2-√3-2√3/32-√3/3
315°7π/4-√2/2√2/2-1-√2√2-1
330°11π/6-1/2√3/2-√3/3-22√3/3-√3

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Evaluate $\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$

In the given trigonometric expression, cosine and sine functions are involved. Two cosine functions with $100$ degrees and $40$ degrees are multiplied and two sine functions with $100$ degrees and $40$ degrees are also multiplied. The summation of them has to evaluate in this trigonometry problem.

$\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$

Simplify the trigonometric expression

The values of cosine of $100$ degrees, cosine of $40$ degrees, sine of $100$ degrees and sine of $40$ degrees are unknown. So, it is recommendable to simplify the given trigonometric expression for evaluating its value. So, let’s analyze or analyse the possibility of simplifying the given trigonometric expression.

$\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$

  1. Two cosine functions get multiplied and two sine functions get multiplied in the given trigonometric expression.
  2. Both cosines and sines functions contain the same angles $100$ degrees and $40$ degrees.
  3. The products of trigonometric functions are connected by a plus sign for forming a trigonometric expression in this problem.

The three notable factors help us to simplify the given trigonometric expression and these factors made the given trigonometric expression to represent the cosine of angle difference trigonometric formula. Hence, we can simplify the given trigonometric expression by the cos angle difference identity.

$\implies$ $\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$ $\,=\,$ $\cos{(100^\circ-40^\circ)}$

We can now simplify the trigonometric expression further by finding the difference between the angles in the cosine function.

$\,\,\,=\,\,\,\,\,\,$ $\cos{(100^\circ-40^\circ)}$

$\,\,\,=\,\,\,\,\,\,$ $\cos{(60^\circ)}$

Evaluate the cosine of angle

The cosine of sixty degrees is known to us and it is equal to the quotient of $1$ by $2$.

$\,\,\,=\,\,\,\,\,\,$ $\dfrac{1}{2}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\cos{100^\circ} \cos{40^\circ}$ $+$ $\sin{100^\circ} \sin{40^\circ}$ $\,=\,$ $\dfrac{1}{2}$

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Cos 40 Degrees

FAQs on Cos 40 Degrees

What is Cos 40 Degrees?

Cos 40 degrees is the value of cosine trigonometric function for an angle equal to 40 degrees. The value of cos 40° is 0.766 (approx)

What is the Value of Cos 40° in Terms of Cosec 40°?

Since the cosine function can be represented using the cosecant function, we can write cos 40° as [√(cosec²(40°) - 1)/cosec 40°]. The value of cosec 40° is equal to 1.55572.

What is the Value of Cos 40 Degrees in Terms of Cot 40°?

We can represent the cosine function in terms of the cotangent function using trig identities, cos 40° can be written as cot 40°/√(1 + cot²(40°)). Here, the value of cot 40° is equal to 1.19175.

How to Find Cos 40° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cos 40° can be given in terms of other trigonometric functions as:

  • ± √(1-sin²(40°))
  • ± 1/√(1 + tan²(40°))
  • ± cot 40°/√(1 + cot²(40°))
  • ± √(cosec²(40°) - 1)/cosec 40°
  • 1/sec 40°

☛ Also check: trigonometry table

How to Find the Value of Cos 40 Degrees?

The value of cos 40 degrees can be calculated by constructing an angle of 40° with the x-axis, and then finding the coordinates of the corresponding point (0.766, 0.6428) on the unit circle. The value of cos 40° is equal to the x-coordinate (0.766). ∴ cos 40° = 0.766.

 

Sours: https://www.cuemath.com/trigonometry/cos-40-degrees/
cos 40 + cos 80 + cos 160 , prediksi UNAS SMA 2016 , DRILL matematika seri 001 no 29

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40 degrees cos

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