formula to find tan 40⁰ and sin 40⁰
Answers
Tan 40 Degrees:-
We can write tan 40o = tan (30 + 10)
we know the formula
tan (A + B) = (tan A + tan B)/ (1 – tan A tan B)
The tangent formula:-
Tan θ = Opposite side/Adjacent side
Tan Values
tan 0° = 0/1 = 0
tan 30° = [(√1/4)/√(3/4)] = 1/√3
tan 45° = 1
tan 60° = [(√3/2)/(½)] = √3
tan 90° = 1/0 = ∞
tan (30 + 10) = (tan 30 + tan 10)/(1 – tan 30 tan 10)
Substitute the approximate value
tan 40° = (0.57735 + 0.176326) /( 1-0.1018)
tan 40° = 0.753676/0.8982
tan 40° = 0.839095
Sin 40 Degrees:-
The value of sin 40 degrees is 0.6427876. . .. Sin 40 degrees in radians is written as sin (40° × π/180°), i.e., sin (2π/9) or sin (0.698131. . .). In this article, we will discuss the methods to find the value of sin 40 degrees with examples.
Sin 40°: 0.6427876. . .
Sin (-40 degrees): -0.6427876. . .
Sin 40° in radians: sin (2π/9) or sin (0.6981317 . . .)
What is the Value of Sin 40 Degrees?
The value of sin 40 degrees in decimal is 0.642787609. . .. Sin 40 degrees can also be expressed using the equivalent of the given angle (40 degrees) in radians (0.69813 . . .).
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 40 degrees = 40° × (π/180°) rad = 2π/9 or 0.6981 . . .
∴ sin 40° = sin(0.6981) = 0.6427876. . .
Explanation:
For sin 40 degrees, the angle 40° lies between 0° and 90° (First Quadrant). Since sine function is positive in the first quadrant, thus sin 40° value = 0.6427876. . .
Since the sine function is a periodic function, we can represent sin 40° as, sin 40 degrees = sin(40° + n × 360°), n ∈ Z.
⇒ sin 40° = sin 400° = sin 760°, and so on.
Note: Since, sine is an odd function, the value of sin(-40°) = -sin(40°).
Methods to Find Value of Sin 40 Degrees
The sine function is positive in the 1st quadrant. The value of sin 40° is given as 0.64278. . .. We can find the value of sin 40 degrees by:
Using Trigonometric Functions
Using Unit Circle
Sin 40° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 40 degrees as:
± √(1-cos²(40°))
± tan 40°/√(1 + tan²(40°))
± 1/√(1 + cot²(40°))
± √(sec²(40°) - 1)/sec 40°
1/cosec 40°
Note: Since 40° lies in the 1st Quadrant, the final value of sin 40° will be positive.
We can use trigonometric identities to represent sin 40° as,
sin(180° - 40°) = sin 140°
-sin(180° + 40°) = -sin 220°
cos(90° - 40°) = cos 50°
-cos(90° + 40°) = -cos 130°
Sin 40 Degrees Using Unit Circle
To find the value of sin 40 degrees using the unit circle:
Rotate ‘r’ anticlockwise to form a 40° angle with the positive x-axis.
The sin of 40 degrees equals the y-coordinate(0.6428) of the point of intersection (0.766, 0.6428) of unit circle and r.
Hence the value of sin 40° = y = 0.6428 (approx)