The values of the six trigonometric functions are:
sin θ = 9/41, cos θ = 40/41, tan θ = 9/40, cot θ = 40/9, sec θ = 41/40, cosec θ = 41/9.
What are trigonometric functions?
The values of all trigonometric functions dependent on the value of the ratio of sides of a right-angled triangle are known as trigonometric ratios. The trigonometric ratios of a right-angled triangle's sides with regard to any of its acute angles are known as that angle's trigonometric ratios.
The three sides of the right-angled triangle are:
Hypotenuse (the longest side)
Perpendicular (opposite side to the angle)
Base (Adjacent side to the angle)
The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).
The trigonometry ratios for a specific angle ‘θ’ is given below:
Trigonometric Ratios:
Sin θ = Perpendicular/Hypotenuse
Cos θ = Base/Hypotenuse
Tan θ = Perpendicular/Base or Sin θ/Cos θ
Cot θ = Base/Perpendicular or 1/tan θ
Sec θ = Hypotenuse/Base or 1/cos θ
Cosec θ = Hypotenuse/Perpendicular or 1/sin θ
What is Pythagoras theorem?
According to the Pythagoras theorem, we can say that in a right-angled triangle:
Hypotenuese² = Base² + Perpendicular²
How do we solve the given question?
We have to find the six trigonometric functions of an angle in standard position if the point with coordinates (40, 9) lies on its terminal side.
With the angle being θ, we have drawn a figure of the case. (attached)
In the right-angled triangle AOB, with respect to angle θ,
Hypotenuse: AO, Perpendicular: AB, and Base: BO
First we derive the value of AO, using the Pythagoras theorem,
AO² = AB² + BO² = 9² + 40² = 81 + 1600 = 1681 = 41²
∴ AO = 41 units.
Now we find the value of the six trigonometric functions, with respect to the angle θ.
sin θ = Perpendicular/Hypotenuse = AB/AO = 9/41
cos θ = Base/Hypotenuse = BO/AO = 40/41
tan θ = sin θ/cos θ = (9/41)/(40/41) = 9/40
cot θ = 1/tan θ = 1/(9/40) = 40/9
sec θ = 1/cos θ = 1/(40/41) = 41/40
cosec θ = 1/sin θ = 1/(9/40) = 40/9.
∴ The values of the six trigonometric functions are:
sin θ = 9/41, cos θ = 40/41, tan θ = 9/40, cot θ = 40/9, sec θ = 41/40, cosec θ = 41/9.
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