How would you find the sine of 30° without using a calculator?
You know about a 30, 60 90 triangle having legs of 1 and sqrt 3 and hypotenuse of 2
so
sin 30 = 1/2
To find the sine of 30° without using a calculator, you can use the special angles or trigonometric ratios.
Step 1: Convert the angle from degrees to radians. Since the sine function typically works with radians, we need to convert 30° to radians. We know that π radians equals 180°, so to convert degrees to radians, we can use the formula: radians = (π/180) * degrees. For 30°, the calculation would be: radians = (π/180) * 30 = π/6.
Step 2: Use the special angle property of sine. The special angle property states that for certain angles (like 30°, 45°, and 60°), the value of the sine can be determined without a calculator. For 30° specifically, the sine value is 1/2.
Therefore, the sine of 30° is 1/2.
To find the sine of 30° without using a calculator, you can use the concept of special right triangles, specifically the 30-60-90 triangle.
In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3 times the length of the smaller side.
Since we know that the sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, we can use this information to find the sine of 30°.
In a 30-60-90 triangle, the hypotenuse is the longest side, and the side opposite the 30-degree angle is the smaller side. Let's assume that the length of the hypotenuse is 2 (you can choose any length for the hypotenuse as it is just a scale factor).
According to the properties of a 30-60-90 triangle, the side opposite the 30-degree angle will be half the length of the hypotenuse, which is 1.
So, the ratio of the side opposite the angle (1) to the hypotenuse (2) gives us the sine of 30°, which is 1/2.
Therefore, the sine of 30° is 0.5.