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Question: How do you find the exact value of \(\tan 245{}^\circ \)?...

How do you find the exact value of tan⁡245∘\tan 245{}^\circ ?

Explanation

Solution

We will rewrite the given angle as a sum of two angles. One of the angles will be a standard angle. We will expand the tangent function of the sum of two angles using the formula for it. We will substitute the value of the trigonometric function of the standard angle. Then we will estimate the value of tan⁡245∘\tan 245{}^\circ and then get the exact value by using a calculator.

Complete step by step answer:
We have to find the value of tan⁡245∘\tan 245{}^\circ . We can write 245 as 245=180+65245=180+65. Therefore, we have
tan⁡245∘=tan⁡(180∘+65∘)\tan 245{}^\circ =\tan \left( 180{}^\circ +65{}^\circ \right)
We have a formula for expanding the tangent function of the sum of two angles. This formula is given as
tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan \left( A+B \right)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}
Now, substituting A=180∘A=180{}^\circ and B=65∘B=65{}^\circ , we get the following expression,
tan⁡(180∘+65∘)=tan⁡180∘+tan⁡65∘1−tan⁡180∘tan⁡65∘\tan \left( 180{}^\circ +65{}^\circ \right)=\dfrac{\tan 180{}^\circ +\tan 65{}^\circ }{1-\tan 180{}^\circ \tan 65{}^\circ }
The angle 180∘180{}^\circ is a standard angle. We know that tan⁡180∘=0\tan 180{}^\circ =0. Substituting this value in the expression above, we get the following,
tan⁡(180∘+65∘)=0+tan⁡65∘1−0×tan⁡65∘ ∴tan⁡(180∘+65∘)=tan⁡65∘ \begin{aligned} & \tan \left( 180{}^\circ +65{}^\circ \right)=\dfrac{0+\tan 65{}^\circ }{1-0\times \tan 65{}^\circ } \\\ & \therefore \tan \left( 180{}^\circ +65{}^\circ \right)=\tan 65{}^\circ \\\ \end{aligned}
Now, 60∘60{}^\circ is also a standard angle and we know that tan⁡60∘=3=1.732\tan 60{}^\circ =\sqrt{3}=1.732. So, we can say that the value of tan⁡65∘\tan 65{}^\circ is greater than 1.732.
Using the calculator, we obtain the exact value as tan⁡65∘=2.145\tan 65{}^\circ =2.145.

Note: We should be familiar with the use of a scientific calculator to obtain values of trigonometric functions for different angles. It is important to know the values of trigonometric functions for standard angles. These values can be used to guess the range of the expected answer. The formula for trigonometric function of sum of two angles is also used to prove some of the identities of trigonometric functions.