Question Video: Using Trigonometric Values of Special Angles to Evaluate Trigonometric Expressions

Find the value of sin (2𝑋/3) tan (4𝑋/3) + cos (2𝑋/3) without using a calculator, given tan 𝑋 = 1, where 𝑋 is an acute angle.

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Video Transcript

Find the value of the sin of two 𝑋 over three multiplied by the tan of four 𝑋 over three plus the cos of two 𝑋 over three without using a calculator, given the tan of 𝑋 equals one, where 𝑋 is an acute angle.

As 𝑋 is an acute angle, we know it lies between zero and 90 degrees. We’ll now use our knowledge of the inverse trigonometric functions together with special angles between zero and 90 degrees to solve this problem. We are told that the tan of 𝑋 is equal to one. Taking the inverse tangent of both sides of this equation, we have the inverse tan of the tan of 𝑋 is equal to the inverse tan of one. We recall that if angle 𝜃 is acute, the inverse tan of tan 𝜃 equals 𝜃. This means that the left-hand side of our equation simplifies to 𝑋. And this is equal to the inverse tan of one.

One of our special angles states that the tan of 45 degrees is equal to one. Taking the inverse tangent of both sides of this equation, we see that 45 degrees is equal to the inverse tan of one. This means that 𝑋 is equal to 45 degrees.

At this stage, it is worth recalling all of our special angles: the sin, cos, and tan of 30, 45, and 60 degrees. The sine of these three angles is equal to one-half, root two over two, and root three over two, respectively. The cosine of the three angles is equal to root three over two, root two over two, and one-half. As the tangent of any angle is equal to the sine of that angle divided by the cosine of that angle, we can calculate the tan of 30 degrees by dividing the sin of 30 degrees by the cos of 30 degrees. One-half divided by root three over two is equal to one over root three. Repeating this process for 45 and 60 degrees, we get answers of one and root three.

We can now use all of this information to calculate the value of the sin of two 𝑋 over three multiplied by the tan of four 𝑋 over three plus the cos of two 𝑋 over three. As 𝑋 is equal to 45 degrees, two 𝑋 over three is equal to 30 degrees as two multiplied by 45 is 90 and dividing this by three gives us 30. Four 𝑋 over three or four-thirds 𝑋 is therefore equal to double this. This is equal to 60 degrees. The expression we need to find the value of is therefore equal to the sin of 30 degrees multiplied by the tan of 60 degrees plus the cos of 30 degrees.

All three of these values appear in our table. They are equal to one-half, root three, and root three over two, respectively. One-half multiplied by root three is equal to root three over two. And we need to add root three over two to this. As the denominators are the same, we simply add the numerators, giving us two root three over two. Dividing the numerator and denominator by two, this simplifies to root three. If the tan of 𝑋 is equal to one where 𝑋 is an acute angle, then the sin of two 𝑋 over three multiplied by the tan of four 𝑋 over three plus the cos of two 𝑋 over three is equal to root three.

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