For small angles θ, the numerical value of sin θ is approximat...

Question

For small angles θ, the numerical value of sin θ is approximately the same as the numerical value of tan θ. Find the largest angle for which sine and tangent agree to within two significant figures.

Accepted Answer

Hello, fellow physicists today we solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. What is the largest angle theta rounded to two significant figures for which the numerical values of cosine theta and cotangent of theta are approximately equal. So that's our end goals. We're trying to figure out what the largest angle for theta when it's rounded to two significant figures for which the numerical values of cosine theta and cotangent of theta are approximately equal to one another. Awesome. We're also given some multiple choice answers. Let's read them off to see what our final answer might be. A is one radian B is 0.79 radiance C is 3.1 radiance and D is 1.6 radiance. Awesome. So first off, in order to solve for this problem, we need to find the largest angle for which cosine of theta and cotangent of theta agree within two significant figures. And we need to determine when the numerical values of cosine theta and cotangent of theta will be nearly equal to one another. So recall that for large angles, cosine, theta would be approximately equal to cotangent of theta. Let us also recall that cotangent of theta is equal to one divided by tangent of theta. And that this is important and that tangent of theta is equal to sine of theta divided by cosine of theta. Awesome. Now let us set cosine of theta of approximately equal to cotangent of theta. So writing that as a mathematical expression, so cosine of theta is approximately equal to which is the little wavy equals sign is equal to cotangent of theta. And now we need to substitute one divided by tangent of theta in for cotangent of theta. So cosine of theta is approximately equal to one divided by tangent of theta. OK. So now we need to substitute sine of theta divided by cosine of theta in for tangent of theta. So cosine of theta is approximately equal to one divided by sine of theta, divided by cosine of theta. Thus, cosine theta is approximately equal to cosine theta divided by sin theta. And this is when we simplify our other equation, it was simplified to this meaning our cosine theta is approximately equal to one divided by sin of theta, divided by cosine theta. So once we simplify that we will get that cosine theta is approximately equal to cosine theta divided by sine theta. OK. Moving right along. So now we need to simplify this equation by multiplying both sides by sine A theta. So cosine a theta multiplied by S A theta is approximately approximately equal to cosine of theta. So this equation applies that and let's make a little star here. This implies that sine of theta is approximately equal to one. So as long as cosine theta cannot equal zero. So basically, just to reiterate. So sine, so we like the equation we wrote which is cosine theta multiplied by sine of theta is approximately equal to cosine theta. So this implies that sin of theta is approximately equal to one. As long as cosine, theta cannot equal zero. Since the division by zero is undefined as it will say, you'll get like a little error message in your calculator if you try to do this. So in order to find theta, we need to find the angle at which sign of theta will be approximately equal to one. So this occurs when theta is very close to 90 degrees or pi divided by two radians. Therefore, the largest angle for which cosine of theta and cotangent of theta agree within two significant figures is approximately, we'll draw a little arrow pi divided by two radians, which is equal to when you divide pi by two in your calculator, you'll get. So pi divided by two is 1.57079. And I'll just go on for a little bit in your calculator, which is approximately equal to 1.6 radians when we rounded two significant figures. So, so one is significant and we have five, which is significant. But as you notice like our decimal right after that is a seven, so that means we can round the five to a six. So that's our final answer. Hooray. So 1.6 radiance is our final answer. So looking at our multiple choice answers, the correct answer has to be D 1.6. Radiance. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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For small angles θ, the numerical value of sin θ is approximately the same as the numerical value of tan θ. Find the largest angle for which sine and tangent agree to within two significant figures.

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