Question Video: Finding the Value of a Trigonometric Function Involving Special Angles

Find the value of cos 2𝑋 without using a calculator, given 𝑋, where tan 𝑋 = 1, is an acute angle.

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Video Transcript

Find the value of the cos of two 𝑋 without using a calculator, given 𝑋, where the tan of 𝑋 equals one, is an acute angle.

We are told that 𝑋 is an acute angle. Therefore, it lies between zero and 90 degrees. We will be able to solve this problem using our inverse trigonometric functions together with our knowledge of special angles. We know that if 𝜃 is an acute angle, the inverse tan of tan 𝜃 is equal to 𝜃. In this question, we are told that the tan of 𝑋 is equal to one. If we take the inverse tangent of both sides of this equation, we get the inverse tan of the tan of 𝑋 is equal to the inverse tan of one.

The left-hand side simplifies to give us 𝑋. Therefore, 𝑋 is equal to the inverse tan of one. Using our knowledge of special angles, we know that the tan of 45 degrees is equal to one. Taking the inverse tangent of both sides of this equation, 45 degrees is equal to the inverse tan of one. We can therefore conclude that the angle 𝑋 is equal to 45 degrees.

We can then use this information to calculate the cos of two 𝑋. As 𝑋 is equal to 45 degrees, two 𝑋 equals 90 degrees. We need to calculate the cos of 90 degrees. The graph of 𝑦 equals cos 𝜃 is as shown. This means that the cos of 90 degrees is equal to zero. If the tan of 𝑋 is equal to one and 𝑋 is an acute angle, then the cos of two 𝑋 equals zero.

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