Question Video: Solving Trigonometric Equations Involving Sum and Difference of Angles Identities | Nagwa Question Video: Solving Trigonometric Equations Involving Sum and Difference of Angles Identities | Nagwa

Question Video: Solving Trigonometric Equations Involving Sum and Difference of Angles Identities

Find the solution set of πœƒ given (tan 25πœƒ βˆ’ tan 23πœƒ)/(1 + tan 25πœƒ tan 23πœƒ) = √3 where 0Β° < ΞΈ < 90Β°.

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Video Transcript

Find the solution set of πœƒ given tan 25πœƒ minus tan 23πœƒ divided by one plus tan 25πœƒ multiplied by tan 23πœƒ equals root three, where πœƒ lies between zero and 90 degrees.

In order to solve this problem, we need to use the compound angle trigonometrical identity. tan of 𝐴 minus 𝐡 is equal to tan 𝐴 minus tan 𝐡 over one plus tan 𝐴 multiplied by tan 𝐡. In our question, the angle 𝐴 is equal to 25πœƒ. And the angle 𝐡 is equal to 23πœƒ. This means that tan of 25πœƒ minus 23πœƒ is equal to root three. 25πœƒ minus 23πœƒ is equal to two πœƒ. Therefore, tan of two πœƒ equals root three.

Taking the inverse tan of both sides of this equation gives us two πœƒ equal tan to the minus one or inverse tan of root three. Inverse tan of root three is equal to 60 degrees. Therefore, two πœƒ is equal to 60. We can therefore find one solution to this problem by dividing both sides of this equation by two.

However, at this stage, it’s important to check if there’re any other solutions within our range, in this case between zero and 90 degrees. We could check this by drawing the tan graph or alternatively using the CAST diagram as shown. There will be one solution in the 𝐴 quadrant and one solution in the 𝑇 for tan quadrant.

We have already calculated that the solution between zero and 90 is 60 degrees. The solution in the 𝑇 quadrant will be 240 degrees as 180 plus 60 equals 240. This means that two πœƒ could be equal to 60 or 240. We only wanted solutions between zero and 90. When we divide 240 by two, our answer would be outside of this range, as 240 divided by two equals 120. This means that there’s only one solution. πœƒ equals 30 degrees as 60 divided by two is equal to 30.

The only solution for tan 25πœƒ minus tan 23πœƒ divided by one plus tan 25πœƒ multiplied by tan 23πœƒ equals root three between zero and 90 is πœƒ equals 30 degrees.

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