![]() ![]() Now, I can write everything in terms of sines and cosines to better understand what’s going on. ![]() Cot(π/2 – x) is equal to tanx, and the cos(π/2 – x) is equal to the sinx. They’re not very difficult to memorize and are useful to know. I recommend either having a reference sheet accessible or refreshing your memory of them on a regular basis. What does π/2 – x represent? These are our co-function identities. Using reciprocal identities, I recognize my answer as sec²θ. Sin²θ + cos²θ = 1, so I can simplify to 1/cos²θ. Recognize the Pythagorean identity in the numerator. Please don’t eliminate the cosines because the division property only works across multiplication, not addition and subtraction. In this example, a lot of students see the squares and get overwhelmed. In the next two examples, we’ll talk about two different identities that make the problems look difficult but are actually fairly simple. When problems look more difficult than they need to be, we can make things simpler by using our identities. My denominator will divide to equal 1 and my simplified answer is cos²θ/sin²θ, which can be simplified to cot²θ. If I do this in the numerator, nothing divides out and I get cos ⋅ cos and sin ⋅ sin. The reciprocal of sinθ/cosθ is cosθ/sinθ. The only way to do this is by multiplying by its reciprocal. We have a big denominator, so if I want to eliminate it, I need to make sure it will equal 1. In the last example, I multiplied by (cos/1) so that the cosines would divide out. Students will often get ‘cancel happy’ and start dividing things out when they see the same terms in the numerator and denominator, but you’ve got to be careful. I’ll start by rewriting this in terms of sines and cosines and then use the same process as the last example. Because everything is being divided by 1, we can write this as sin²θ. In the numerator, cosine is over 1, so I now have cosθ and cosθ that will divide to equal one. ![]() In the denominator, everything will multiply to equal 1. What can I multiply by that will divide with cosθ? In this case, cosθ is in the denominator, so I’ll multiply by cosθ on the top and the bottom to produce equivalent fractions. We’ll use the division property to eliminate fractions. To get rid of cosθ, I want to multiply by secθ. Sin²θ is divided by cosine, and the whole numerator is divided by 1/cosθ. Unfortunately, nothing simplifies out, but we can simplify the numerator to a fraction by multiplying sinθ by sinθ, giving us sin²θ. This step brought in more fractions, which we typically want to eliminate, but we also have an additional numerator and some denominators. In this example, the first thing I’ll do is rewrite tangent as sin/cos and secant as 1/cos. When I have a rational expression with the tangent or functions that aren’t in terms of sines and cosines, it’s easiest to rewrite the terms to see what can be divided out or what operations you can apply. In the next examples, we’ll add an extra step as we divide out terms. I’ll use parentheses around (sinx/cosine), which, using the quotient identity, I know simplifies to tanx. It doesn’t matter if it’s in the numerator, the denominator, or just in front. Sometimes students see the negative in the numerator and get stuck, but we can put the negative in front as a product. By applying even-odd identities, we can see that sin(-x) is equivalent to -sinx, and the cosine(-x) is equivalent to the positive cosine(x). Think even-odd identities whenever you see a negative angle inside the argument of a trigonometric function. The first thing I notice here is the even-odd identities. Using my reciprocal identity, I can simplify to tanθ. I’m left with 1 as the numerator, and my denominator is still cotθ. According to the division property, whenever you have the same term in the numerator and the denominator separated by multiplication, they’ll divide to equal one. I’ll also reference it in other examples. Now is a great time to review the division property, so I’ll include a picture of it written out. In this example, I want to rewrite cosθsecθ in terms of sines and cosines so they’ll be reciprocals of one another, so their product will equal one. As we solve each problem, we’ll look to the division property, which I’ll explain and reference in almost every example. These types of problems can be confusing because a rational expression is generally a fraction, and a lot of students struggle with fractions. We’ll look at how to reduce terms down to a single number or a single trigonometric function. Today, I’ll go over eight examples of simplifying trig expressions using division. ![]()
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