Sin 0
In trigonometry, there are three major or primary function, Sine, Cosine and Tangent, which are used to find the angles and length of the right-angled triangle. Before discussing Sin 0, let us know about Sine function.
Sine function defines a relation between the angle and perpendicular side and hypotenuse side. Or you can sin theta (the angle formed between the hypotenuse and adjacent side) is equal to the ratio of perpendicular and hypotenuse of a right-angled triangle.
Let us discuss more of the trigonometric sine functions here in this article.
Sine Definition In Terms of Sin 0
As we have already discussed, the sin of angle theta is a ratio of the length of the opposite side, perpendicular and hypotenuse of the right-angled triangle.
Sin \(\theta\) = \(\frac{Opposite Side}{Hypotenuse}\)
= \(\frac{Perpendicular}{Hypotenuse}\)
Now if we want to calculate sin 0 degrees value, we have to check the coordinates point on x and y plane. Sin 0 signifies that the value of x coordinate is 1 and the value of y coordinate is 0,i.e. (x,y) is (1,0). That means the value of the opposite side or perpendicular is zero and the value of hypotenuse is 1. So if we place the values in sin ratio for \(\theta\)=0^{0} , perpendicular side= 1 and hypotenuse as 0, then we get,
Sin 0^{0 }=0/1
Or
Sin 0^{0 }= 0
From the above equation, we have yield sin 0 degrees value. Now let us write other sin degrees or radians values for one full revolution, in a table.
Sine Degrees/Radians | Values |
Sin 0^{0} | 0 |
Sin 30^{0 }or Sin π/6 | 1/2 |
Sin 45^{0} or Sin π/4 | \(1/\sqrt{2}\) |
Sin 60^{0 }or Sin π/3 | \(\sqrt{3}/2\) |
Sin 90^{0} or Sin π/2 | 1 |
Sin 180°c or Sin π | 0 |
Sin 270° or Sin 3π/2 | -1 |
Sin 360° or Sin 2π | 0 |
If we write opposite of the value of Sin degrees, we get the values of cos degrees. Because, Sin \(\theta\)=1/Cos\(\theta\)
Therefore we can write,
Sin 0^{0}= Cos 90^{0}=0
Sin 30^{0}=Cos 60^{0}=½
Sin 45^{0}=Cos 45^{0} = \(1/\sqrt{2}\)
Sin 60^{0}=Cos 30^{0}=\(\sqrt{3}/2\)
Sin 90^{0}=Cos 0^{0}=1
In the same way, we can write the values for Tan degrees.
Tan \(\theta\)=Sin\(\theta\)/Cos\(\theta\)
Therefore,
Tan 0^{0}=Sin 0^{0}/Cos 0^{0}=0
Tan 30^{0}=Sin 30^{0}/Cos 30^{0}=\(\sqrt{3}\)/2
Tan 45^{0}=Sin 45^{0}/Cos 45^{0}=1
Tan 60^{0}=Sin 60^{0}/Cos 60^{0}=\(\sqrt{3}\)
Tan 90^{0}=Sin 90^{0}/Cos 90^{0}=Undefined
We learned about sin theta 0 degrees value along with other degree values here, this far. Also, derived the value for cos degree and tan degrees with respect sin degrees. In the same way, we can find other trigonometric ratios like sec, cosec and cot.
Based on these values, we can draw the trigonometry table,
Angle | 0^{0} | 30^{0} | 45^{0} | 60^{0} | 90^{0} | 180^{0} | 270^{0} | 360^{0} |
Sin A | 0 | 1/2 | \(1/\sqrt{2}\) | \(\sqrt{3}/2\) | 1 | 0 | -1 | 0 |
Cos A | 1 | \(\sqrt{3}/2\) | \(1/\sqrt{2}\) | 1/2 | 0 | -1 | 0 | 1 |
Tan A | 0 | 1/\(\sqrt{3}\) | 1 | \(\sqrt{3}\) | Undefined | 0 | Undefined | 0 |
Example: Find the value of Sin 90^{0}+Cos 90^{0}
Solution: As we know, Sin 90^{0}=1
And Cos 90^{0}=0
Therefore, Sin 90^{0}+Cos 90^{0 }= 1+0 = 1
Example: Find the value of Sin 270°+2Tan 45^{0}
Solution: Sin 270°= -1
And Tan 45^{0}=1
Therefore, Sin 270°+2Tan 45^{0 }= -1+2*1 = -1+2 = 1
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MATHS Related Links | |
Trigonometric Ratios of Complementary Angles | Trigonometric Functions |
Trigonometric Ratios Of Standard Angles | Trigonometric Equations |