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01.14.2019: One Lesson of Math - Special Edition: A Review of Trigonometry


Today's soundtrack is Stephen Cleobury's arrangement of John Rutter: Requiem, a gorgeous, spiritual, choral album. I first heard its main theme performed by my sister-in-law and her musical theater class, and it's stuck with me since.


This afternoon, I'm working on the assignment portion of the "Angles in Standard Position" lesson that I've been learning. First sessions: Part 1, Part 2.


I did a lot of review of how trigonometry works to get through this assignment. I've written up something of a trigonometry cheat sheet to help myself out in the future. Here's what I've reviewed:

 

Before we begin...


There are three major trigonometric ratios: Sine, Cosine, and Tangent. They are represented on the calculator as SIN, COS, and TAN. By using the mnemonic "SOH CAH TOA," we can remember that SIN = Opposite over Hypotenuse, COS = Adjacent over Hypotenuse, and TAN = Opposite over Adjacent. This refers to the ratios of the sides in relation to the angle we are looking at, often called θ, or "theta."


When we look at right triangles, we can use the Pythagorean theorem to determine the length of one side of a triangle if we know two of the other sides. The Pythagorean theorem tells us that a² + b² = c², with c representing the hypotenuse (longest side) of the triangle.


Right triangles will also always have a combined total of 180°.


Between SOH CAH TOA, the 180° rule, and the Pythagorean theorem, we can "solve" (find all angles and lengths of) any right triangle with a minimum of information. We can solve an entire triangle as long as we know either the length of one side and the measure of one angle or the lengths of two sides.


Here are some examples:

 

How to find the length of all sides when we know the lengths of two sides


If we know the length of two sides of a right triangle, we first apply the Pythagorean theorem, a² + b² = c².


If one of my sides is 4" and the hypotenuse is 7", I'll need to find out how long the third side is.


I'll let 4 represent a and 7 represent c.


Already I have my equation sorted: 4² + b² = 7², which comes to 16 + b² = 49.


Now I'll isolate my variable by subtracting 16 from both sides of the equation: b² = 33.


So my third side is the square root of 33, which we can represent by simply writing √33.


So we now know that the lengths of the three sides of the triangle are 4", 7", and √33" (which is approximately 5.7").

 

How to find the lengths of all sides when we know one angle and the length of one side


If we know the measure of one angle (in degrees) and the length of one side, we can find the lengths of all sides of our triangle by using trigonometry.


Let's say for example that I know that one of my angles is 49°, and the length of my hypotenuse is 3'. I want to find out how long the other two legs are.


In this example, my known angle is on the bottom left side of this triangle: ◢. I'll call the lower leg x, and the vertical leg on the right will be y. So I know the length of my hypotenuse, and I need to find the length of my opposite side (x) and my adjacent side (y).


When choosing a trigonometric ratio, we need to choose one that includes our known angle - in this case, the hypotenuse. I'm going to use COS and SIN, since, as we know from SOH CAH TOA, SIN will include Opposite and Hypotenuse, and COS will include Adjacent and Hypotenuse.


Let's look for the length of x first. x is the adjacent side, so we will be working with COS first. We know that the cosine (or COS) of an angle (θ) is equal to the length of the adjacent over (or "/") the length of the hypotenuse. So let's write it that way now: COS θ = a/h.


Now that we've identified which ratio we'll be using, we can fill in our known information. So in COS θ = a/h, we can insert our known degrees (49) and known length (h = 3), and we'll call a "x", as we discussed earlier. This gives us COS 49 = x/3.


Now that we have our equation, we need to isolate the variable so that we can solve it! Using basic algebra, we can simply multiply both sides of our equation, COS 49 = x/3, by 3. This will isolate x. We now have 3 COS 49 = x.


And now...It's Calculator time.


In my calculator, I type in 3[cos](49, and it gives me 1.968177087, which rounds nicely to 1.97. So the length of x is 1.97'.


To find the length of y, I have two options: I can either follow the same steps as above but use SIN instead of COS, or I can use the Pythagorean theorem. Either way is fine.

 

How to find the measure of an angle when we know the measure of one other acute angle


If we know the measure of one acute angle and we wish to find the measure of the other, we can simply use the 180° rule: all three angles will add up to 180°. So since we know that our right angle is 90°, we can take 180-90-[known angle] and we will get our unknown angle's measure.


For example, if I know that the measure of one angle is 49° like in our example above, I would say 180-90 is 90; 90-49 is 41. So my unknown angle has to be 41°.

 

How to find the measure of an angle when we know the length of two sides


Time to whip out that calculator again. When we know the length of any two sides of a right triangle and want to find the measure of any acute angle in the triangle, we need to first find what the relationship between the sides and the angle is. Are they opposite/hypotenuse? adjacent/hypotenuse? or adjacent/opposite? If the first, we will use SIN; if the second, COS; if the third, we will use TAN.


Now, when it comes to finding angles, we are actually reverse engineering what we did a couple of examples up, where we found a side based on an angle. This time, we're finding an angle based on sides, so we're going to use the inverse trigonometric ratio, which will be shown on most calculators as [ratio]⁻¹. On my calculator (a Texas Instruments TI-83 Plus), I use the yellow [2nd] function key to access these ratios.


Let's look at another triangle, the same shape as before: ◢. I want to find the angle at the bottom left. I'll call it θ. I know that the vertical leg is 8mm long, and the horizontal leg is 5mm long. The horizontal leg is adjacent to θ, and the vertical leg is opposite to θ. So since I know the length of the opposite leg and the adjacent, I can use SOH CAH TOA to recognize that I'll be using the inverse of TAN to find this angle.


Now, we'll be multiplying TAN⁻¹ by the opposite over the adjacent. So let's put that into an equation: TAN⁻¹ (8/5) = θ.


When we type TAN⁻¹ (8/5) into the calculator, it gives us 57.99. To the nearest degree, this gives us θ = 58°.

 

So there it is! By merely knowing one length and one angle, or by knowing two lengths, we can solve any right triangle.


Hope this review of trigonometry helps you as much as it helped me! I've put together a downloadable version of this guide; you can find it here.

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