In Exercises 23–34, find the exact value of each of the remain...

Question

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = 4/3, cos θ < 0

Accepted Answer

Hello everybody. I'm everything night today. Today we're going to begin this question, that state determine the value of the five other trigonometric ratios. And the one that we are given is tangent theta is equal to 35 divided by 12. And they also let us know that cosine theta is less than zero. So when they tell us that 10 data is equal to 35 divided by 12, we have to ask ourselves, what does that mean? What is tangent data? Well, tangent theta is equal to sign of theta divided by cosine of the. So we know that tangent theta will be positive in two quadrants in a graph, specifically a quadrant in which both sine and cosine are positive or in which they are both negative. So therefore, tangent data will be positive in quadrants one and three because in those quadrants sine data and cosine data are both the same sign. So this is important because we're going to be, we have to discover what quadrant we are in so that we can do a small graph that will help us determine the size of each side and we can go ahead and solve for the ratios needed. So for cosine theta, they don't tell us the value, but they tell us that it's less than zero and cosine data data is negative in quadrants two and three. So w which quadrant do they both have in common have quadrant three in common. Therefore, we will be in quadrant three. Now, before moving on, let's recall an I very important rule when it comes to trigonometric ratios and that is the rule of silk to now this is an acronym that I'll show you how it works now. So the first letter in each word so is S stands for sign, the C and C stands for cosine and the T and to stands for tangent, the O stands for opposite and the, the A stands for adjacent and the H stands for hypotenuse. And you just imagine like for example, we'll do to because we are, they tell us tangents. So we'll have that tangent data is equal to, we said that the O means opposite. So we have tangent data is equal to opposites and the A stands for adjacent. So we have opposite and imagine that in between the opposite, the second two letters of the word to you are dividing the last two letters so have opposite, divided by the A which is adjacent, which means that the length of the opposite side of theta will be 35 the adjacent side of theta will be 12. So this is important because now we can go ahead and start a graph where we focus on quadrant three and we draw a hypo coming from the origin into quadrant three and then a straight line down to create a right triangle with the X axis. Now we said that the opposite side of theta and we'll label theta as the angle quadrant three. We said that the opposite side of theta is 35. So we labeled that as 35 the adjacent side of theta is 12. Now we'll label the hypos as C because what do we use when we have a right triangle? And we need to figure out the measurements of the sides. We use the Pythagorean theorem which is A squared plus B squared equals C squared A and B being the two legs of the triangle and C being the hypos. So we have 12 squared plus 35 squared equals C squared or 144 plus 1225 is equal to C squared. When you add the two numbers up on the left hand side, we'll have 1369 is equal to C squared. And if we take the square root of both sides, we'll have that C is equal to 37. So now we have the measurement of the hypotenuse and we can move on to the other trigonometric ratios. Now, something that we should note is we are in quadrant three, which will affect the sign, the sign of our trigonometric ratios. So if it's positive or negative. So in quadrant three tangent and cold tangents are positive, anything else will be negative so that it's, that's something important to know before we move on to the ratios. And now that we have noted that let's move on. So let's start with sign of data sin of theta if we use. So we have S S which is sine theta, we'll focus on the. So, so we have sine theta is equal to the O which is opposite, divided by the second letter which is H standing for a hypotenuse. So we have opposite divided by the hypotenuse. So we have 35 divided by the hypotenuse 37. And remember that this will be negative because we are in quadrant three. Now cosine data, we look at the car. So we have cosine is equal to the A which is, which is adjacent, divided by the second letter which is an H hypotenuse. So we have adjacent, which is 12 divided by hypotenuse, which is 37. And once again, this will be negative because we're in quadrate three and cosine is negative in quadrate three, cosine, cosine is negative in quadrants two and three as we mentioned earlier. Now let's move on to co sequent theta, we'll skip tangent because they already gave us tangent beta. So sequent data in quadrant three will be negative. So we have negative one divided by sine theta. So code sequence is just one divided by the sine. So we have cos theta equals negative one divided by sin theta which is negative. And we just take the reciprocal of the sine data that we calculated. So we have negative 37 divided by and we can move on to second data in quadrant three will be negative and we'll have negative one divided by cosine theta. So we once again take the reciprocal of cosine now and we'll have negative 37 divided by 12. And finally, we look at co tangent data which as we said earlier is positive in quarter three. So we have cotangent data is one divided by 10 tangent data and they gave us 10 data earlier as 35 divided by 12. So the reciprocal of that will be 12 divided by 35. And just like that, we have all six trigger the magic ratios where they gave us the tangents and we were able to solve for the other five ratios using this small graph and some calculations. So I hope that was helpful. And until next time.

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = 4/3, cos θ < 0

Key Concepts

Understanding Trigonometric Ratios

Sign of Trigonometric Functions in Quadrants

Using the Pythagorean Identity

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