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Question: Find whether the value of the expression \( \left( \sin 80{}^\circ -\cos 80{}^\circ \right) \) is ne...

Find whether the value of the expression (sin⁡80∘−cos⁡80∘)\left( \sin 80{}^\circ -\cos 80{}^\circ \right) is negative or positive.

Explanation

Solution

Hint : First we memorize the trigonometric table of values. Trigonometric table is associated with a right-angled triangle. We observe the values of sine and cosine function at different trigonometric angles and give the answer to this question.

Complete step-by-step answer :
We have given the expression (sin⁡80∘−cos⁡80∘)\left( \sin 80{}^\circ -\cos 80{}^\circ \right) .
We have to find that the value of a given expression is negative or positive.
Now, let us memorize the values of trigonometric ratios. In the table we discuss the sine, cosine and tangent function values at standard angle values.

Angle (In degree)sin⁡θ\sin \thetacos⁡θ\cos \thetatan⁡θ\tan \theta
0∘0{}^\circ001100
30∘30{}^\circ12\dfrac{1}{2}32\dfrac{\sqrt{3}}{2}13\dfrac{1}{\sqrt{3}}
45∘45{}^\circ12\dfrac{1}{\sqrt{2}}12\dfrac{1}{\sqrt{2}}11
60∘60{}^\circ32\dfrac{\sqrt{3}}{2}12\dfrac{1}{2}3\sqrt{3}
90∘90{}^\circ1100∞\infty

Now, when we analyze the values of sine and cosine functions, we observe that the value of sine function is increasing as the value of angle increases and value of cosine function decreases as the value of angle increases, till 45 degree the value will be same and hence the difference will be zero and after that it keeps on increasing as the value of sin function is more compared to cosine. So, when we subtract the value of the cosine function from the value of the sine function we will get a positive value.comes and zero and after that it keeps on increasing as the value of sin function is more compared to cosine.
Let us understand by an example.
If we have an expression like (sin⁡90∘−cos⁡90∘)\left( \sin 90{}^\circ -\cos 90{}^\circ \right)
Then, the value of expression will be 1−0=11-0=1 , which is a positive integer.
So, the value of the expression (sin⁡80∘−cos⁡80∘)\left( \sin 80{}^\circ -\cos 80{}^\circ \right) is also positive.
Note : Alternatively we directly solve the given expression by using a scientific calculator. We will find the value of (sin⁡80∘−cos⁡80∘)\left( \sin 80{}^\circ -\cos 80{}^\circ \right) and give the answer whether it is positive or negative. Here we use a trigonometric ratio table to find the answer. Trigonometric ratio tables help us to solve problems easily. Trigonometric tables are used in engineering, navigation science, geometry, geography etc.