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In this video, we’re going to see how to apply the inverse of the three trigonometric ratios, sine, cosine, and tangent, in order to calculate angles in right-angled triangles.
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First of all, let’s define these inverse trigonometric ratios.
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I have here a diagram of a right-angled triangle in which I’ve labeled one of the angles as 𝜃.
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I’ve then labeled the three sides of the triangle in relation to the angle 𝜃, so we have the opposite, the adjacent, and the hypotenuse.
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The three trigonometric ratios, sine, cosine, and tangent, are the ratios that exist between different pairs of sides in this right-angled triangle.
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So, the sine ratio, sin of angle 𝜃, is the opposite divided by the hypotenuse.
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Cosine ratio is cos of 𝜃 is the adjacent divided by the hypotenuse.
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And finally, tangent, the tan ratio, is the opposite divided by the adjacent.
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A useful way to remember these is to recall the word SOHCAHTOA, where each of those letters represents the first letter in each of these words.
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So, CAH, for example, the C is for cos, the A is for adjacent, and the H is for hypotenuse.
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So, CAH tells us that the cos ratio is adjacent divided by hypotenuse.
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Those ratios are the trigonometric ratios, and they’re particularly useful if we know a side and an angle and are looking to calculate another side.
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But in this video, we’re looking at how to calculate angles.
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And therefore, we need what’s referred to as the inverse trigonometric ratios.
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These are defined like this.
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The notation we use is sin, cos, or tan and then a superscript negative one, which is said as sine inverse or inverse sine.
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And what they mean are, if I know the value of the ratio of opposite divided by hypotenuse, then I can work backwards using this inverse sine function in order to calculate the angle that that ratio is associated with.
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So, when we know two sides of a right-angled triangle, we can use the relevant inverse trigonometric ratio in order to calculate an angle.
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On your calculator, you will see that above the sin, cos, and tan button, there is usually sin inverse, cos inverse, and tan inverse.
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You often have to press shift in order to get to these functions.
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We’ll now look at a couple of examples of how to apply these inverse trigonometric ratios.
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For the given figure, find the measure of angle 𝜃 in degrees to two decimal places.
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So, we have a diagram of a right-angled triangle, and we can see that we’re given the lengths of two of the sides of this triangle.
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They’re three units and eight units.
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And we’re looking to find the size of this angle 𝜃.
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As we’re using trigonometry for this problem, the first step is going to be to label all three of the sides of the triangle in relation to their angle 𝜃.
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So, we have the opposite, the adjacent, and the hypotenuse.
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We can see then that the two sides of the triangle we have are the adjacent and the hypotenuse.
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If I think back to the acronym of SOHCAHTOA, then A and H appear together in the CAH part, which tells me that it’s the cosine ratio I’m going to need to use in this question.
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The definition of the cosine ratio, remember, was that cos of the angle 𝜃 is equal to the adjacent divided by the hypotenuse.
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So, I’m gonna write down this ratio using the information in this particular question.
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And therefore, I have that cos of 𝜃 is equal to three over eight.
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Now, this is where I need to use the inverse trigonometric function.
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If cos of 𝜃 is equal to three over eight, then 𝜃 is equal to cos inverse of three over eight.
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At this stage, I use my calculator to evaluate this, remembering that that cos inverse button is usually directly above the cos button.
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This tells me that 𝜃 is equal to 67.97568 degrees.
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The question asked me to round my answer to two decimal places.
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Therefore, my final answer is that 𝜃 is equal to 67.98 degrees.
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So, in this question, we identified the need for the cosine ratio because the lengths we were given were the adjacent and the hypotenuse.
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We wrote down that ratio using these lengths.
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We then used the inverse cosine ratio in order to calculate the value of this angle 𝜃.
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Find the measure of angle ACB giving the answer to the nearest second.
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I’ve got a diagram of a right-angled triangle, and I’m asked to find the measure of angle ACB.
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So, that means the angle formed when I move from A to C to B.
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It’s this angle here that I’m looking for, so I’ve given it the label 𝜃.
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As we’re going to use trigonometry to solve this problem, I’m gonna label the three sides of the triangle with their names in relation to that angle of 𝜃.
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So, we have the opposite, the adjacent, and the hypotenuse.
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Now, I can see that the two sides I’ve been given are the opposite and the adjacent.
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If I think back to that acronym SOHCAHTOA, then I see that it’s the tan ratio I’m going to need because the opposite and the adjacent appear together in the TOA part.
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So, the definition of the tan ratio is the opposite divided by the adjacent.
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For this triangle, then, that is 43 divided by 26.
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So, we have tan of 𝜃 is equal to 43 over 26.
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Now, I need to use the inverse tangent ratio.
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So, if tan 𝜃 is 43 over 26, then 𝜃 is equal to tan inverse of 43 over 26.
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I use my calculator to evaluate this.
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And it tells me that 𝜃 is equal to 58.84069.
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Now, this is an answer in degrees, and the question has asked me to give my answer to the nearest second.
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So, I need to recall how to convert a value from degrees into degrees, minutes, and seconds.
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So, I have 58 full degrees, and then I have this decimal of 0.840695 and so on, which needs to be converted into minutes and then seconds.
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Remember that a minute is one sixtieth of a degree.
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So, to work out what this decimal represents in minutes, I need to multiply it by 60.
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When I do that, I get this value of 50.4417295.
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This tells me that there are 50 full minutes and a decimal of 0.4417295, which needs to be converted into seconds.
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A second is one sixtieth of a minute.
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So, again, to convert this into seconds, I need to multiply it by 60.
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When I do this, I get a value of 26.5037.
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So, in order to round this to the nearest second, I round it up to 27 seconds.
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Finally, I need to pull the three parts of this answer together.
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And in doing so, my final answer then is that the measure of angle ACB is 58 degrees, 50 minutes, 27 seconds to the nearest second.
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So, within this question, we identified the need for the tan ratio, because the lengths we were given were the opposite and the adjacent.
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We wrote down the tan ratio for this triangle.
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We then applied the inverse tan ratio to work out the angle that this ratio was associated with and finally converted this answer from degrees into degrees, minutes, and seconds.
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A car is going down a ramp which is 10 metres high and 71 metres long.
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Find the angle between the ramp and the horizontal, giving the answer to the nearest second.
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So, this question is a worded problem, and we haven’t been given a diagram.
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I would always suggest that, in this situation, you draw your own diagram to start off with.
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So, here we have an idea of what this ramp might look like.
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It is 71 metres long and 10 metres high.
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We’re asked to find the angle between the ramp and the horizontal, so we’re looking to calculate this angle here, which I’ve labeled as 𝜃.
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Now, we’re going to solve this problem using trigonometry.
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So, I’m going to begin by labeling the three sides of this triangle in relation to this angle 𝜃.
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Having done this, I can see that the two lengths that I’ve been given represent the opposite and the hypotenuse of this right-angled triangle.
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Thinking back to that acronym of SOHCAHTOA, this tells me that it’s the sine ratio I’m going to need in this problem, as O and H appear together in the SOH part of SOHCAHTOA.
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So, the definition of the sine ratio is that it’s the opposite divided by the hypotenuse.
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I then write this ratio down for this particular triangle, and I have then that sin of 𝜃 is equal to 10 over 71.
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In order to work out the size of this angle, I need to apply the inverse sine ratio.
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So, if sin of 𝜃 is 10 over 71, then 𝜃 is sin inverse of 10 over 71.
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I evaluate this with my calculator, and I see that 𝜃 is equal to 8.09674977.
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Now, this is an answer in degrees, and the question has asked me to give my answer to the nearest second.
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So, let’s recall how to convert an answer from degrees into degrees, minutes, and seconds.
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I have eight full degrees, and then I have a decimal left over of 0.09674977, which needs to be converted first into minutes and then into seconds.
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Now, recall that a minute is one sixtieth of a degree.
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So, in order to work out what this decimal represents in minutes, I need to multiply it by 60.
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This gives me a value of 5.804986188.
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What this tells me then is that I have five full minutes and then a decimal of 0.80 and so on left over.
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This decimal needs to be converted into seconds.
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So, recall that a second is one sixtieth of a minute.
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And therefore, in order to see what this decimal represents in seconds, I need to multiply it by 60.
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When I do this, I get a value of 48.29917126.
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And if I round that to the nearest second, it’s 48 seconds.
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Finally, then, I need to combine the three parts of my answer.
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And this tells me that the angle between the ramp and the horizontal, to the nearest second, is eight degrees, five minutes, and 48 seconds.
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So, within this question, we drew our own diagram as we weren’t given one within the question itself.
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We identified the need for the sine ratio because the two lengths we have been given were the lengths of the opposite and the hypotenuse.
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We used the inverse sine ratio to calculate the angle that was associated with that value.
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And then, we converted our answer from degrees into degrees, minutes, and seconds.
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In summary, then, the three inverse trigonometric ratios sine inverse, cosine inverse, and tan inverse can be used to calculate an angle in a right-angled triangle when we know at least two of the sides.
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The ratio that we choose will depend on which two sides we’re given in exactly the same way as it does when we’re calculating the length.