Question Video: Evaluation a Trigonometric Expression Involving Double-Angle Identities given a Trigonometric Function of an Angle Mathematics • 10th Grade

Video Transcript

Find the value of one plus sin two 𝐴 divided by one plus cos two 𝐴 given the tan of 𝐴 is five twenty-sixths where 𝐴 is greater than zero and less than 𝜋 over three radians without using a calculator.

Before trying to calculate the value of the expression, let’s consider the information we are given. We are told that the tan of angle 𝐴 is five twenty-sixths and that 𝐴 lies between zero and 𝜋 by three radians. Using the CAST diagram, we know that if an angle lies between zero and 𝜋 by two, the sine of the angle, the cosine of the angle, and the tangent of the angle will all be positive. This is true in this case as the angle 𝐴 lies between zero and 𝜋 by three radians.

Using our knowledge of right-angle trigonometry, we know that the sin of angle 𝜃 is equal to the opposite over the hypotenuse, the cos of angle 𝜃 is equal to the adjacent over the hypotenuse, and the tan of angle 𝜃 is equal to the opposite over the adjacent. In this question, we are told the tan of angle 𝐴 is equal to five twenty-sixths. We can calculate the length of the hypotenuse using the Pythagorean theorem. ℎ squared is equal to 26 squared plus five squared. 26 squared plus five squared is equal to 701. We can then square root both sides of our equation such that ℎ is equal to the square root of 701. The sin of angle 𝐴 is, therefore, equal to five over the square root of 701 and the cos of angle 𝐴 is equal to 26 over the square root of 701.

Let’s now consider the expression one plus the sin of two 𝐴 divided by one plus the cos of two 𝐴. We know that the sin of two 𝜃 is equal two sin 𝜃 cos 𝜃 and that the cos of two 𝜃 is equal to two multiplied by cos squared 𝜃 minus one. The expression in this question can be rewritten as one plus two sin 𝐴 cos 𝐴 divided by one plus two cos squared 𝐴 minus one. On the denominator, one plus one is equal to zero.

We can now substitute the values of sin 𝐴 and cos 𝐴 into this expression. On the numerator, we have one plus two multiplied by five over the square root of 701 multiplied by 26 over the square root of 701. On the denominator, we have two multiplied by 26 over the square root of 701 squared. The numerator simplifies to 961 over 701 and the denominator simplifies to 1352 over 701. Multiplying the numerator and denominator by 701 gives us a final answer of 961 over 1352. This is the value of one plus sin two 𝐴 divided by one plus cos two 𝐴.