Solving For Side Lengths and Angles: Learning to Utilize Trigonometry

Now that we have established a basic definition for trigonometry and explored the meaning behind its 3 ratios, we can use it to solve problems and find a triangle's angles and side lengths. As long as we know the value of 1 side length and 1 angle (not the right angle) in a right triangle, we are able to solve for the triangle's missing values.

If you will remember from the previous blog, all of the ratios have equations. These are sinθ = opp/hyp, cosθ = adj/hyp, and tanθ = opp/adj. All we need to do for any given right triangle is to substitute the values and then algebraically solve for them.

In the triangle below, we can see that we know the value of 1 of the triangle's side lengths as well as one of the triangle's angles (other than the right angle). Now, we need to decide which ratio to use. Since we have the value of an angle, and we have the value of the side length which is adjacent to the angle, we must use a ratio that utilizes the adjacent side. This immediately knocks out the possibility of using sine as sine requires both the opposite side and the hypotenuse, both of which we do not have the value of. This leaves us with 2 possibilities, either cosine or tangent. Depending on which side we want to find first, the ratio which is used will differ. For our example, we are going to try and solve for the opposite side first.

(softschools.com, 2020)

Since we know that the formula for finding the opposite side, in this case, is tangent, we start off by writing the equation and substituting all the necessary values.

tanθ = opp/adj

(Starting equation)

tan25° = b/8

(Substituting in all the values, it is fine to remove the unit of measurement, such as m for 8, for ease of calculation, but don't forget to include it in your final answer)

Now, we can algebraically solve this equation by isolating the unknown variable, b. This would look like this:

b = (tan25)(8)

(Multiplying both sides by 8 to isolate b)

Now, on a calculator, you want to find the button which has tan printed on it. Then, you want to type tan25 into your calculator. Then, you want to multiply the answer of tan25 by 8, giving you a final value for b, and telling you the side length of the opposite side.

b ≈ 3.73 m

(Remember to include your unit of measurement after calculations)

After finding the value for side b, we can now finish the perimeter of the triangle by solving for side c. Once again, we start by deciding on which ratio we want to use. Now that we have the values of both the adjacent and opposite sides, we can use any ratio that utilizes the hypotenuse. This instantly eliminates tangent from being a possibility and leaves us with the option of using either sine or cosine. Both of which are useable in this scenario. We are going to use cosine as the value of the adjacent side is a whole number, making it easier to calculate. However, it is important to note that both ratios, in this case, would work, and you will get the same answer if you used sine.

Since we know that the ratio that we are using in this scenario is cosine, let's start off by writing the equation.

cosθ = adj/hyp

(Starting equation)

cos25° = 8/c

(Substituting in the values)

And now we can algebraically solve this equation like we did in the example prior.

(cos25)(c) = 8

c = 8/cos25

(Isolating the variable, c)

c ≈ 8.83 m

It is also important to note that you do not have to use trigonometry to find the side length of an undefined side if you already know the values of the other 2 sides. We can simply use the Pythagorean Theorem to solve for side c. Since we have already learned that the Pythagorean Theorem is a^2 + b^2 = c^2, all we need to do is substitute and solve once again.

a^2 + b^2 = c^2

8^2 + 3.73^2 = c^2

77.91 = c^2

√77.91 = √c^2

8.83 ≈ c

c ≈ 8.83 m

Now that we have determined all the side values and the total perimeter of the triangle, all that is left to completely solve this triangle is by finding its undefined angle. Since we know all of the side lengths on the triangle already, we can use any of the 3 ratios to find the value of the final angle. Let's pick a random one and use sine. Now, all we have to do is to repeat the steps we used above. Remember that the opposite side is different because the reference angle changed, the hypotenuse, however, will always stay the same. Also remember, that since we are solving for an angle, we have to utilize arcsine, which can be seen below.

sinθ = opp/hyp

sinθ = 8/8.83

θ = sin^-1(8/8.83)

θ ≈ 65°

Another way that we can find the final angle is to understand the simple properties of a triangle. Within a triangle, the internal angles always add up to 180°. No more, no less. Therefore, if we already have 2 out of the 3 total angles, all we need to do is to subtract the angles from the total. This would look like this:

θ = 180° - 90° - 25°

θ = 65°

Now that we know that the final angle of the triangle is 65°, we have fully solved the triangle. Good job! Now you know how to solve a right triangle using trigonometry. However, how do you solve a different triangle? One that does not have a right angle? We will explore that question in the next blog.

softschools.com. (2020). Using Trig Ratios to Solve Triangles: Sides. In softschools.com. https://www.softschools.com/math/calculus/using_trig_ratios_to_solve_triangles_sides/