What is the period of 𝑓 of 𝑥 equals two-thirds cosine of three-fourths 𝑥?
The general period formula is equal to the regular period of the function, so whatever the function is and in this case it’s cosine, divided by the absolute value of 𝐵. And 𝐵 is the constant that is being multiplied to the variable on the inside of that parenthesis. So here, 𝐵 would be the three-fourths that’s being multiplied to the 𝑥 on the inside of that function. So what is the regular period of the cosine function? That is two 𝜋. And then as we said, 𝐵 is equal to three-fourths. And it’s the absolute value of three-fourths, which is three-fourths because the absolute value sign makes anything positive. So since it’s a positive three-fourths, it stays at three-fourths.
So we can go ahead and take what we have but change two 𝜋 to be two 𝜋 over one. So here we are dividing fractions. And when you divide fractions, we end up multiplying by the denominator’s reciprocal. So we take our top fraction on the numerator, two 𝜋 divided by one, and instead of dividing by three-fourths, we are multiplying by the reciprocal of three-fourths. So we’re actually multiplying by four-thirds. So we take two 𝜋 times four, which is eight 𝜋. And on the denominator, one times three is three.
Therefore, the period of this function would be eight 𝜋-thirds.