in this lesson we are going to introduce you to trigonometry in grade 10 this will always involve a right-angled triangle in trigonometry people like to use Greek letters to show angles here we use the Greek letter theta in any right-angle triangle the side opposite the right angle is called the hypotenuse also note this hypotenuse lies next to the angle theta on the other side of theta side adjacent to theta and the third side is the opposite side because it lies opposite theta let's recap the hypotenuse always lies opposite the right angle the adjacent side lies next to theta and the opposite side lies opposite theta this will always be true no matter in what position the triangle lies there are three important ratios in this triangle the first is opposite over hypotenuse next is adjacent over hypotenuse and the third is opposite over adjacent these three ratios have special names the first one is called sine of theta or just sine theta the next ratio is called cosine of theta or just cos theta and the third ratio is called tangent of theta or just tan theta this information is extremely important in trigonometry let's go on to see how to remember it the order is always sine cos tan or SCT then you need to remember oh heck another hour of algebra and from now on I will use the letters Oh a and H for opposite adjacent and hypotenuse these are the first letters in oh heck another hour of algebra please note it does not matter how big or small the triangles are if the angles are the same the ratios will always have the same value in other words sine theta will have the same value in both triangles cos theta will have the same value and so will tan theta let's go on to see how to use all of this information here is the first example we are given a triangle and are asked to find sine theta cos theta and tan theta step one is to identify the size of the triangle the hypotenuse is easy because it is the side that lies opposite the right angle the side next to theta is the adjacent side the side opposite theta is the opposite side next we need our three ratios remember the order a sine cos tan and then oh heck another hour of algebra the first ratio we are asked to find is sine theta which is opposite over hypotenuse from the triangle we see that opposite is sex and hypotenuse is 10 so the value of the ratio sine theta is 0 comma sex this completes part 1 of the question in part two we are asked to find the ratio cos theta first we need to recall that cos is adjacent over hypotenuse using the triangle we see that this is 8 over 10 or 0 comma 8 this completes part 2 of the question in part three we are asked to find the ratio tan theta first we need to recall that tan is opposite over adjacent using the information on the triangle this is 6 over 8 which gives an answer of zero comma seven five this completes the question click pause if you need more time to check the working here is the next example we are given another triangle and are asked to find the same three trig ratios again we begin by identifying the sides hypotenuse lies opposite the right angle adjacent lies next to theta and opposite lies opposite theta but we have a problem can you see what it is the length of the opposite side has not been given now what because this is a right angle triangle we can use Pythagoras to find the length of oh we can write that o squared plus 15 squared equals 17 squared which means that o is equal to the square root of 17 squared minus 15 squared this gives an answer of eight which we write in on the triangle we are now ready to find the values of the three trig ratios to help us remember them we write down sign cos and tan and then oh heck another hour of algebra this reminds us that sine theta is the ratio opposite over hypotenuse which in this case is 8 over 17 8/17 is equal to zero comma five correct to one decimal place this completes part 1 of the question in part two we are asked to find the value of cos theta first we need to recall that causes the ratio adjacent over hypotenuse which in this case is 15 over 17 15/17 is equal to zero comma nine correct to one decimal place this completes part two of the question in part three we are asked to find the value of tan theta first we need to recall that tan is the ratio opposite over adjacent which in this case is 8 over 15 8/15 is equal to zero comma five correct to one decimal place this completes the question in grade 10 you will also be required to do trigonometry on the Cartesian plane here we are given point P a point in the first quadrant we can use this point to draw a right angle triangle to do this we draw a line from P perpendicular to the x-axis the length of the side on the x-axis is X which we get from the x coordinate of point P and the length of the side parallel to the y-axis is y which we get from the y coordinate of point P the side which joins the origin to P is always labeled R the angle which is always measured from the x-axis to R is Theta our is the hypotenuse because it lies opposite the right angle X is the site adjacent to theta and why is the opposite theta the three trig ratios are sine cos and tan together with oh heck another hour of algebra this means that sine theta is the ratio Y of R which is the ratio opposite over hypotenuse and cos theta is X of R which is the ratio adjacent over hypotenuse and tan theta is y of X which is the ratio opposite over adjacent these are the ratios you need to use when doing trigonometry on the Cartesian plane let's see how to use these ratios if they give us the point three four first we complete a right-angle triangle by drawing the following line perpendicular to the x-axis the x coordinate of P is the length of the side on x-axis the y coordinate of P gives the length of the side parallel to the y-axis to find the length of our the hypotenuse we again need to use Pythagoras the length of our is therefore 5 we are now ready to find three trig ratios sine theta equals four over five or zero comma eight cos theta equals 3 over 5 or 0 comma sex and tan theta equals 4 over 3 or 1 comma 3 correct to one decimal place it is now time to taste yourself good luck