Question Video: Evaluating Trigonometric Expressions Involving Special Angles

Find the value of 3 cos 2π‘₯ tan 2π‘₯ without using a calculator, given sin π‘₯ = tan 30Β° sin 60Β° where π‘₯ is an acute angle.

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Video Transcript

Find the value of three multiplied by the cos of two π‘₯ multiplied by the tan of two π‘₯ without using a calculator, given the sin of π‘₯ is equal to the tan of 30 degrees multiplied by the sin of 60 degrees where π‘₯ is an acute angle.

In order to answer this question, we will need to recall the inverse trigonometric functions together with our knowledge of special angles. We are told that π‘₯ is an acute angle, which means that it lies between zero and 90 degrees. As π‘₯ lies between these values, it is worth recalling our special angle values: the sin, cos, and tan of 30, 45, 60 degrees. The sin of 30 degrees is equal to one-half, the sin of 45 degrees is root two over two, and the sin of 60 degrees is root three over two. The cos of these three angles is equal to root three over two, root two over two, and one-half, respectively. The tan of 30 degrees is equal to one over root three, which is equivalent to root three over three. The tangent of 45 degrees is equal to one, and the tangent of 60 degrees is root three.

We recall that the tan of angle πœƒ is equal to the sin of πœƒ divided by the cos of πœƒ. This means that we could calculate the values in the bottom row by dividing the sine values by the corresponding cosine values. For example, root three over two divided by a half is equal to root three. Now that we have recalled these special angles, let’s consider what else we are told in the question.

We know that the sin of angle π‘₯ is equal to the tan of 30 degrees multiplied by the sin of 60 degrees. The tan of 30 degrees is equal to one over root three, and the sin of 60 degrees is equal to root three over two. The sin of angle π‘₯ is, therefore, equal to one over root three multiplied by root three over two. Multiplying the two values on the right-hand side gives us one-half. The sin of angle π‘₯ is, therefore, equal to one-half.

At this stage, we recall from our inverse trigonometric functions that, for any acute angle πœƒ, the inverse sin of sin πœƒ is equal to πœƒ. We can, therefore, take the inverse sine of both sides of our equation. Since π‘₯ is an acute angle, π‘₯ is, therefore, equal to the inverse sin of one-half. From our table, we see that the sin of 30 degrees equals one-half. Taking the inverse sine of both sides of this equation, we see that 30 degrees is equal to the inverse sin of one-half. This means that π‘₯ is equal to 30 degrees.

We will now clear some space to calculate the value of three multiplied by the cos of two π‘₯ multiplied by the tan of two π‘₯. If π‘₯ is equal to 30 degrees, two π‘₯ must be equal to 60 degrees as two multiplied by 30 is 60. Our expression can, therefore, be rewritten as three multiplied by the cos of 60 degrees multiplied by the tan of 60 degrees. Once again, we can find these values from our table. The cos of 60 degrees equals one-half, and the tan of 60 degrees equals root three. Our expression simplifies to three multiplied by a half multiplied by root three. This is equal to three root three over two.

Alternatively, we might have noticed that the cos of two π‘₯ multiplied by the tan of two π‘₯ is equal to a different trigonometric function. As the tan of πœƒ is equal to the sin of πœƒ divided by the cos of πœƒ, multiplying both sides by the cos of πœƒ, we see that the cos of πœƒ multiplied by the tan of πœƒ is equal to the sin of πœƒ. This means that the cos of two π‘₯ multiplied by the tan of two π‘₯ is equal to the sin of two π‘₯. We can, therefore, simply calculate three multiplied by the sin of 60 degrees. As the sin of 60 degrees is equal to root three over two, multiplying this by three gives us three root three over two.

The value of three multiplied by the cos of two π‘₯ multiplied by the tan of two π‘₯ where π‘₯ is an acute angle and sin π‘₯ is equal to the tan of 30 degrees multiplied by the sin of 60 degrees is three root three over two.

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