Question Video: Using Double Angle Identities to Evaluate a Trigonometric Expression Mathematics

Evaluate (36 tan 112°30′)/(1 − tan² 112°30′) without using a calculator.

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Video Transcript

Evaluate 36 multiplied by the tan of 112 degrees, 30 minutes divided by one minus tan squared of 112 degrees, 30 minutes without using a calculator.

In order to answer this question, we begin by recalling one of our double-angle identities. This states that the tan of two 𝜃 is equal to two multiplied by the tan of 𝜃 divided by one minus tan squared 𝜃. We could multiply both sides of this equation by 18 such that 18 multiplied by tan two 𝜃 is equal to 36 tan 𝜃 divided by one minus tan squared 𝜃. The right-hand side of our equation is now written in a similar form to the expression in the question where 𝜃 is equal to 112 degrees and 30 minutes. Doubling both sides of this equation, we see that two 𝜃 is equal to 225 degrees. We can therefore rewrite the initial expression as 18 multiplied the tan of 225 degrees.

We could simply type this into our calculator; however, in this question, we are not allowed to do so. Instead, we will recall one of our special angles. The tan of 45 degrees is equal to one. Using the CAST diagram, we see that 225 degrees lies in the third quadrant. The tan of any angle in this quadrant is positive. As 225 degrees is equal to 180 degrees plus 45 degrees, we know that the tan of 225 degrees will be equal to the tan of 45 degrees. The tan of 225 degrees is equal to one. So our expression simplifies to 18 multiplied by one. This is equal to 18. So 36 multiplied by the tan of 112 degrees and 30 minutes divided by one minus tan squared of 112 degrees and 30 minutes is equal to 18.

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