Understanding the Value of Sine for Special Angles

In trigonometry, the sine of special angles such as 60° and 75° plays a crucial role in various mathematical and real-world applications. This article will delve into the methods of calculating and understanding these values, focusing on the unit circle and the special right triangles technique.

The Sine of 60 Degrees: A Comprehensive Look

Firstly, let's understand the sine of 60°. The sine of 60° can be determined through the unit circle or by using a 30-60-90 triangle. In the context of a 30-60-90 triangle, where the angles are 30°, 60°, and 90°, and the sides are in the ratio 1:√3:2, the sine of 60° is defined as the ratio of the opposite side to the hypotenuse. Therefore, sin 60° √3/2.

It is important to note that the value of sin 60° is positive as it is in the first quadrant, where both sine and cosine functions are positive. The value sin 60° 0.866 is often used in calculations and can be converted to radians thus: sin 60° √3/2 ≈ 0.866, or sin(π/3).

The Sine of 75 Degrees: A Formulaic Approach

For the sine of 75°, one can use the addition formula for sine, which states that sin(a b) sin a cos b cos a sin b. Therefore, sin 75° sin(45° 30°) sin 45° cos 30° cos 45° sin 30°.

Substituting the known values, we get

sin 75° (√2/2) * (√3/2) (√2/2) * (1/2) (√3 1) / (2√2) ≈ 0.965925826.

Using the Unit Circle for Sine Calculations

The unit circle is a powerful tool for determining trigonometric values. For an angle of 60°, the sine value represented by the y-coordinate of the point on the unit circle is √3/2. This is because the coordinates at 60° are (cos 60°, sin 60°) (1/2, √3/2).

For -60°, the sine value remains the same in magnitude but is negative, as sine is an odd function, thus sin(-60°) -√3/2 ≈ -0.8660254.

What is the Value of 'sinsin 60°'?

The expression ‘sinsin 60°’ is grammatically and mathematically incorrect. The sine of 60° is a specific value, approximately 0.866, and not an angle or another function to be evaluated further. Therefore, it is not possible to compute the sine of a sine value without a specific function or context. The same applies to any sine function with an angle that is not specified.

Converting between Degrees and Radians

The measure of an arc can often be confusing when left unmentioned. The measure 0.866 could refer to degrees, in which case the sine of 60° would still apply as 0.866, or it could refer to radians, meaning we would evaluate sin(0.866). The correct interpretation depends on the context, such as whether the problem specifies units or not.

Summary

In summary, the sine of 60° is a standard angle in trigonometry with a known value of √3/2. Similarly, the sine of 75° can be calculated using trigonometric formulas. Using the unit circle, we can easily find these values. The expression 'sinsin 60°' is mathematically incorrect and does not have a meaningful value unless specified otherwise. When dealing with trigonometric functions, it is essential to specify the correct units (degrees or radians).

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