Sin Theta Equals 0 properties is here. Check Sine Definition in terms of Sin 0 value. Know the various Sine degrees and radians along with the formulae, tricks, and tips. Follow sin θ equals zero examples, frequently asked questions, steps to solve trigonometric equations, analysis of the solution, etc. Get the steps to solve trigonometric problems, formulae, examples, solutions, etc.
Do Read:
- Problems on Trigonometric Identities
- Trigonometrical Ratios of 90 Degree Minus Theta
- Trigonometrical Ratios of 180 Degree Minus Theta
Sin 0 Definition
In trigonometric equations, there are 3 primary functions which are sine, cosine, and tangent. These functions are used to calculate the length and angles of the right-angled triangles. The sine function is something that defines the relationship between the hypotenuse side and the angle of the perpendicular side (or) sin θ is defined as the ratio of the hypotenuse and the perpendicular of the right-angled triangle.
Sin θ Formula
If we have to calculate the degree of sin 0 value, then find the coordinates points on the x and y plane. Sin 0 defines the x value where coordinates are 1 and the y coordinates value is 0, which is (x,y) = (1,0) which means that the value of the perpendicular or opposite side is 0 and the hypotenuse value is 1. Therefore, to place the sin ratio values for where θ=00 hypotenuse is 0 and perpendicular side is 1
Sin 0° = 0
or
Sin 0° =0/1
The relations of various trigonometric functions are
sin(θ) = Opposite/Hypotenues
tan(θ) = Opposite/Adjacent
cos(θ) = Adjacent/Hypotenues
From the above-written equations, sin 0 degrees value. Now have a look at radians or degree values for each revolution in the given table.
| Sine Radians / Degrees | Sin Values |
|---|---|
| Sin (0°) | 0 |
| Sin (30°) or Sin (Π/6) | 1/2 |
| Sin (45°) or Sin (Π/4) | 1/√2 |
| Sin (60°) or Sin (Π/3) | 3/√2 |
| Sin (90°) or Sin (Π/2) | 1 |
| Sin (180°) or Sin (Π) | 0 |
| Sin (270°) or Sin (3Π/2) | -1 |
| Sin (360°) or Sin (2Π) | 0 |
As mentioned in the above table, we can determine the values of tan values
Tan(θ) = Sin(θ)/Cos(θ)
Hence,
Tan(0°)=Sin(0°)/Cos(0°) = 0
Tan(30°)=Sin(30°)/Cos(30°) = 3/√2
Tan(450)=Sin(45°)/Cos(45°) = 1
Tan(60°)=Sin(60°)/Cos(60°) = √3
Tan(90°)=Sin(90°)/Cos(90°) = Undefined
2Π is the period for both cosine and sine function. To find all the possible solutions, add 2Πk, where k is an integer to the initial solution. The period of the function is 2Π which states all the possible solutions for the given function.
The equation with the period 2Π for the function is
sinθ = sin(θ ± 2kπ)
For other trigonometric functions also, the possible solutions are indicated by the same rules. To solve the trigonometric equations, we must follow the same techniques that we use for the algebraic equations. We read and write a trigonometric equation from left to right, in the same way as we read the sentence. To make the straightforward process, we must look for the factors, patterns, find the common denominators, the substitution of certain expressions with the variable.
How to solve a Trigonometric Equation?
- First of all, check for the pattern which helps you in minimizing the equation. Mostly the pattern will be of algebraic properties like factoring or a squares opportunity.
- Now, use the single variable and substitute it in the trigonometric equation in such a way that u or x.
- Follow the same pattern of the algebraic equation to solve trigonometric expressions.
- Then, substitute the trigonometric expression in the resultant expression by using the variable.
- Finally, solve the equation to find the angle of the equation.
Table of Trigonometric Ratios for Various Angles
| Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| Angles (In Radians) | 0° | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ | 0 | ∞ | 0 |
| cot | ∞ | √3 | 1 | 1/√3 | 0 | ∞ | 0 | ∞ |
| cosec | ∞ | 2 | √2 | √2/3 | 1 | ∞ | -1 | ∞ |
| sec | 1 | √2/3 | √2 | 2 | ∞ | -1 | ∞ | 1 |
Table of Trigonometric Ratios for Various Radians
| Angle | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| Radian | 0 | Π/6 | Π/4 | Π/3 | Π/2 | Π | 3Π/2 | 2Π |
Applications of Trigonometry
- Trigonometric equations help us to find the missing sides and angles of the triangle.
- These equations are mostly used by builders to measure the distance and height of the building from the viewpoint.
- It is used by the students to solve trigonometry-based problems.
Problems on Sin Theta Equals 0
Problem 1:
If √3 sinθ- cosθ = 0 and 0 < θ < 90°, find the value of θ?
Solution:
As given in the question,
The equation is √3 sinθ- cosθ = 0
√3 sin θ = cos θ
sin θ = cos θ * 1/√3
sin θ / cos θ = 1/√3
tan θ = Tan 30
θ = 30°
Therefore, the value of θ is 30º
Problem 2:
If secθ.sinθ = 0, then find the value of θ?
Solution:
As given in the question,
The equation is secθ.sinθ = 0
As we know that sec θ = 1/cos θ
The equation will be
1/cos θ . sin θ = 0
tan θ = 0
tan θ = tan 45°
θ = 45°
Problem 3:
Find the values of θ in [0°,360) so that y/r = sin θ = 1/2?
Hint: Take y=1,r=2
Solution:
As given in the question,
y/r = sin θ = 1/2
r = 2, y = 1
From the given values, we use the hypotenuse theorem
Hence, we have to find the values of x
i.e., x = √3
Therefore, θ = 30°
As the side of the triangle is not mentioned, there is also another chance where the x can be negative
Hence, if the value of x is negative, then x = -√3
Therefore, θ = 150º
Thus, the values of θ in (0°,360) are 30° and 150°