Why is Sin 0 Equal to 0 Unit Circle Proof and Examples
The formal definition of the Sin function can be expressed in the ratio form of lengths of the opposite side and hypotenuse of any given triangle. In fact, the Sine function is written as when the angle of a right triangle is given as zero grades. In a zero Degree right triangle, the length of the opposite side is zero.
Trigonometry is a branch of mathematics and a sub-branch in algebra concerned with the measurement of specific functions of angles and their application to calculations. An example of Trigonometry which is easy to understand is that of what architects use to calculate any particular distances.
Algebra and Trigonometry are two major branches of mathematics. Algebra involves the study of math with specific formulas, rules, equations, and other variables. Trigonometry deals only with the triangles and their measurements.
Main Functions of an Angle
The six main functions of an angle that are commonly used in Trigonometry are
Sine (Sin),
coSine (cos),
tangent (tan),
cotangent (cot),
secant (sec), and
cosecant (CSC).
What is the Value of Sin 0
A little about Trigonometry and Trigonometric Ratios-
Trigonometric ratios in Trigonometry are derived from the three sides of a right-angled triangle, basically the hypotenuse, the base (adjacent), and the perpendicular (opposite).
According to the trigonometric ratio in maths, there are three basic or primary trigonometric ratios also known as trigonometric identities.
To be more specific, they are used in right-angled triangles, the triangles with one angle equal to 90 Degrees
Sin 0 Value-
\[Sin 0^{^{\circ}} Value = \theta\]
Here are the primary trigonometric functions!
The primary trigonometric functions used are coSine, Sine, and tangent. Sin 0 Degrees value and other trigonometric ratios are used for common angles like 0°, 30°, 45°, 60°, 90° are used in trigonometric equations and calculations.
A Little Information About What a Right-Angled Triangle is?
Let us consider a right-angle triangle named ABC, with its three sides namely the opposite, adjacent, and the hypotenuse. In a right-angled triangle, we generally refer to the three sides according to their relation with the angle. The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°.
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The 3 sides of a right-angled triangle are as follows-
The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse (H).
The side that is opposite to the angle θ is known as the opposite (O).
The side which lies next to the angle is known as the Adjacent(A)
Pythagoras theorem states that,
In a right-angled triangle,
\[(Opposite)^{2}+(Adjacent)2^{2}= (Hypotenuse)^{2}\]
As our angle of interest is Sin 0. So accordingly, the Sin function of an angle or Sin 0 Degrees is equal to the ratio of the length of the opposite side to the length of the hypotenuse (longest side).
Sin 0 Formula -
Sin 0 Value = Opposite side/Hypotenuse side
Derivation of Sin 0 Value
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If we want to calculate the value of Sin 0 Degrees, we need to check the coordinates points on the x and y plane. Sin 0 signifies that the value of x coordinate is equal to 1 and the value of y coordinate is equal to 0, that is the coordinates (x, y) is (1, 0) which means that the value of When we place the values in Sin ratio for θ=0°, substituting perpendicular side= 1 and hypotenuse = 0,
We get,
Sin 0⁰ = 0/1
Or
Sin 0⁰ = 0
Now when we write the opposite of the value of Sin Degrees, we get the values of cos Degrees.
As we know, Sin θ = 1/Cos θ
Therefore, we can now write the Sin and cos values from different angles.
Similarly, we can write the value of Tan Degrees, we get the values of cos and Sin Degrees.
As we know, that the value of Tan θ = Sin θ / Cos θ
Therefore,
Table Showing the Different Values of Sin, Cos, and Tan for Different Angles.
Table Showing Different Values of Sin, Cos, and Tan With Respect to Radians.
Questions to be Solved -
1. Evaluate the value of Sin 90⁰ + Cos 90⁰.
Sol. As we know that the value of Sin 90⁰ = 1
And the value of Cos 90⁰ = 0
Substituting the values of Sin 90⁰ and Cos 90⁰ ,
Therefore, Sin 90⁰ + Cos 90⁰ = 1 + 0
= 1
2. What is the value of Sin 270⁰ + 2 Tan 45⁰.
Sol. As we know that the value of Sin 270⁰ = -1
And the value of Tan 45⁰ = 1
Substituting the values of Sin 270⁰ and Tan 45⁰ ,
Therefore, Sin 270⁰ + 2Tan 45⁰ = -1+2×1 = -1+2 = 1
Fun Facts
Sin inverse is denoted as Sin-1 and it can also be written as arcSin or a Sine
Hipparchus is known as the Father of Trigonometry. The value of arc and chord for a series of angles was discovered by him
Applications of Trigonometry you need to know-
Trigonometric ratios help us to find missing angles and misSing sides of a triangle.
Trigonometry is widely used by the builders to measure the height and distance of the building from its viewpoint.
It is also used by the students to solve the questions based on Trigonometry.
Quiz Time
1. If x and y are considered as complementary angles, then which of the following is correct?
a. Tan x =Tan y
b. Sin x=Sin y
c. Cos x= Cos y
d. Sec x= Cosec y
Ans: Option d
2. What will be the value of Sin 150°?
1/5
2
1/2
1
Ans: Option c
Sin 150°= Sin (90°+60°)
=Cos 60° {Since, (90+θ) = Cosθ}
=1/2
An easy and simple way to learn and understand Trigonometry is by studying all the basics of trigonometric angles and formulas by writing them down in a separate notebook which will be really useful to revise them before exams. Make sure you understand and study all the entire right-angle triangle concepts well so that you might compare any problems with a triangle before you try to solve them. The main thumb rule to score well in Trigonometry is to learn your Pythagoras theorem with a whole heart. Keeping the Sine rule and CoSine rule at your fingertips will help you solve any type of problem in the examination. Finally, list down all the important identities and formulas of Trigonometry in your mind and revision notes as well, and be thorough. Remember to learn how to use the Trigonometry table in the necessary place.
FAQs on Understanding the Value of Sin 0 in Trigonometry
1. What is the value of sin 0?
The value of sin 0 is 0. In trigonometry, sine represents the ratio of the opposite side to the hypotenuse in a right triangle. When the angle is 0°, there is no opposite side length, so the ratio becomes 0. On the unit circle, the coordinate at 0° is (1, 0), and the sine value corresponds to the y-coordinate, which is 0.
2. Why is sin 0 equal to 0?
sin 0 = 0 because there is no vertical (opposite) side when the angle is 0 degrees. In a right triangle, sine is defined as opposite ÷ hypotenuse. At 0°, the triangle collapses along the base, making the opposite side length 0. Using the unit circle definition, the point at 0° is (1, 0), and since sine equals the y-coordinate, the value is 0.
3. How do you prove that sin 0 = 0 using the unit circle?
You can prove sin 0 = 0 using the unit circle because sine equals the y-coordinate of a point on the circle.
- On the unit circle, the radius is 1.
- At 0 radians (or 0°), the point on the circle is (1, 0).
- The sine of an angle equals the y-coordinate.
4. What is sin 0 in radians?
The value of sin 0 radians is 0. Whether the angle is measured in degrees (0°) or radians (0), the sine of zero remains the same. Since 0 radians represents no rotation from the positive x-axis, the sine value (y-coordinate on the unit circle) is 0.
5. What is the exact value of sin 0?
The exact value of sin 0 is 0. Unlike other trigonometric values that may involve square roots or fractions, sin 0 is a whole number. It does not require approximation or rounding because it is already an exact integer value.
6. What is sin 0 on the unit circle?
On the unit circle, sin 0 equals the y-coordinate of the point at 0°, which is 0.
- The unit circle has radius 1.
- The point at 0° is (1, 0).
- Sine corresponds to the vertical coordinate.
7. Is sin 0 positive or negative?
sin 0 is neither positive nor negative; it is exactly 0. Zero is neutral and does not have a sign. On the unit circle, the point (1, 0) lies on the x-axis, so the sine value (y-coordinate) is exactly 0, meaning it is not positive or negative.
8. What is the derivative of sin x at x = 0?
The derivative of sin x at x = 0 is 1. Since the derivative of sin x is cos x, we evaluate cos 0. Because cos 0 = 1, the slope of the sine curve at x = 0 is 1.
9. What is the limit of sin x as x approaches 0?
The limit of sin x as x approaches 0 is 0. Mathematically, lim(x→0) sin x = 0. As x gets closer to 0 from either side, the sine value also approaches 0 continuously. This follows from the continuity of the sine function.
10. What is the value of sin 0 in a right triangle?
In a right triangle, sin 0 = 0 because the opposite side length is 0. Using the sine formula sin θ = opposite / hypotenuse:
- At θ = 0°, the opposite side has length 0.
- The hypotenuse has a positive length.
