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Question: What is the exact value of \[\sin {60^ \circ } - \cos {60^ \circ }\]?...

What is the exact value of sin⁡60∘−cos⁡60∘\sin {60^ \circ } - \cos {60^ \circ }?

Explanation

Solution

To find the exact value of the given function, at first, we will find the value of sin⁡60∘\sin {60^ \circ }and cos⁡60∘\cos {60^ \circ }separately. Then using these values we can find the final value of the given function.

Complete step-by-step solution:
It is given that; the function is sin⁡60∘−cos⁡60∘\sin {60^ \circ } - \cos {60^ \circ }.
We have to find the value of sin⁡60∘−cos⁡60∘\sin {60^ \circ } - \cos {60^ \circ }.
We know that the value of sin⁡60∘\sin {60^ \circ }is 32\dfrac{{\sqrt 3 }}{2}.
Again, the value of cos⁡60∘\cos {60^ \circ }is 12\dfrac{1}{2}.
Substitute the values we get,
sin⁡60∘−cos⁡60∘=32−12\sin {60^ \circ } - \cos {60^ \circ } = \dfrac{{\sqrt 3 }}{2} - \dfrac{1}{2}
Simplifying we get,
sin⁡60∘−cos⁡60∘=3−12\sin {60^ \circ } - \cos {60^ \circ } = \dfrac{{\sqrt 3 - 1}}{2}
Hence, the value of sin⁡60∘−cos⁡60∘\sin {60^ \circ } - \cos {60^ \circ }is 3−12\dfrac{{\sqrt 3 - 1}}{2}.

Note: The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan.
The trigonometry angles which are commonly used in trigonometry problems are 0°, 30°, 45°, 60° and 90°. The trigonometric ratios such as sine, cosine and tangent of these angles are easy to memorize. We will also show the table where all the ratios and their respective angle’s values are mentioned. To find these angles we have to draw a right-angled triangle, in which one of the acute angles will be the corresponding trigonometry angle. These angles will be defined with respect to the ratio associated with it.
There are many real-life examples where trigonometry is used broadly.
If we have been given with height of the building and the angle formed when an object is seen from the top of the building, then the distance between object and bottom of the building can be determined by using the tangent function, such as tan of angle is equal to the ratio of the height of the building and the distance.
Applications of Trigonometry

Its applications are in various fields like oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, etc.
It is also helpful to measure the height of the mountain, find the distance of long rivers, etc.