Question Video: Using the Sum of Angle Identities to Solve Trigonometric Equations Modeling Real-Life Situations Mathematics

The intensity of an electric current is given by 𝐶 = (11/2) sin (105°𝑡), where 𝑡 is the time in seconds. Rewrite the intensity after one second using sum and product formulae in terms of special angles.

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Video Transcript

The intensity of an electric current is given by 𝐶 is equal to 11 over two sin of 105 degrees multiplied by 𝑡, where 𝑡 is the time in seconds. Rewrite the intensity after one second, using sum and product formulae in terms of special angles.

In this question, we are given a formula that calculates the intensity of an electric current. We are asked to rewrite this formula when 𝑡 is equal to one as we want the intensity after one second. Substituting 𝑡 equals one into our formula gives us 𝐶 is equal to 11 over two multiplied by sin of 105 degrees. We could simply evaluate this using our calculator. However, we are asked to rewrite the formula in terms of special angles. The special angles are usually considered to be zero, 30, 45, 60, and 90 degrees. We are expected to recall the sine, cosine, and tangent of these angles. However, this will not be required in this question.

We notice that our angle 105 degrees is equal to 45 degrees plus 60 degrees. This means that we could rewrite the formula as 11 over two multiplied by the sin of 45 degrees plus 60 degrees. Next, we recall that one of the sum of angle identities states that the sin of 𝛼 plus 𝛽 is equal to sin 𝛼 cos 𝛽 plus cos 𝛼 sin 𝛽. Letting 𝛼 be 45 degrees and 𝛽 60 degrees, we have the sin of 45 degrees plus 60 degrees is equal to the sin of 45 degrees multiplied by cos of 60 degrees plus cos 45 degrees multiplied by sin of 60 degrees.

The left-hand side can be rewritten as the sin of 105 degrees. And multiplying through by 11 over two, we have an expression for 11 over two sin of 105 degrees. This is equal to 11 over two multiplied by the sin of 45 degrees cos of 60 degrees plus the cos of 45 degrees sin of 60 degrees.

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