Solving expressions using 30-60-90 special right triangles
All in One PlaceEverything you need for JC, LC, and college level maths and science classes. | Learn with EaseWe’ve mastered the national curriculum so that you can revise with confidence. | Instant Help24/7 access to the best tips, walkthroughs, and practice exercises available. |
Make math click 🤔 and get better grades! 💯Join for Free
Intros
Examples
Lessons
Free to Join!
Easily See Your Progress
We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.Make Use of Our Learning Aids
Earn Achievements as You Learn
Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.Create and Customize Your Avatar
Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
Topic Notes
What is a 30-60-90 triangle?
One of the two special right triangles you'll be facing in trigonometry is the 30-60-90 triangle. The other one is the 45 45 90 triangle. These triangles are special triangles because the ratio of their sides are known to us so we can make use of this information to help us in right triangle trigonometry problems.
In the case of the 30-60-90 triangle, their side's ratios are 1 : 2 : 3. We'll prove that this is true first so that you can more easily remember the triangle's properties.
A 30-60-90 triangle is actually half of an equilateral triangle.
Triangle ABC shown here is an equilateral triangle. Since it's equilateral, each of its 3 angles are 60 degrees respectively. Its sides are also equal.
If we draw a line AD down the middle to bisect angle A into two 30 degree angles, you can now see that the two new triangles inside our original triangle are 30-60-90 triangles.
We know that BD is equalled to DC, which is also equalled to half of AB since AB was originally equalled to BC before it was split in half. So therefore, the ratio of BD : AB is 1 : 2.
Since we have a hypotenuse due to the 90 degree angle that exists inside our triangle, we can use the pythagorean theorem to figure out the last side: AD.
12+AD2=22
AD2=4−1
AD2=3
AD=3
And there you have it! You now know how all the sides of the 30-60-90 triangles came about. Let's move on to solving right triangles with our knowledge on the sides' ratios.
Solve for 30 60 90 triangle
Here we have a 30-60-90 special right triangle, with the three interior angles of 30, 60, 90 degrees. We know that the length of each side in this triangle is in a fixed ratio. We can now use the ratio to solve the following problem.
Question 1:
If Θ = 30° , find the exact value of the following expression:
sin 2Θ
Solution:
Since Θ = 30°, sin 2Θ = sin 60°.
We know that sin Θ = hypotenuseopposite
Since we are looking for the value of sin 60°, we are going to look at the 60° angle in the special triangle and find out the opposite side and hypotenuse of the angle. And we found that the opposite side = 3 and the hypotenuse = 2. Then, we put the numbers into sin 60° = hypotenuseopposite. This gives us the final answer of 23
Question 2:
If Θ=60° , find the exact value of the following expression:
4sin32θ
Solution:
First, we put Θ = 60° into 4sin32θ. We get
4sin3(260) = 4sin3(30°)
From the special triangle, we know that sin 30° = 21
Then, we can now solve the cubic root:
4(21)3=4(81)
The final answer is 21
A good online resource to help you check your work when it comes to 30-60-90 triangles (or even 45 45 90 triangles!) can be found here. You can also play along with this online scalable 30-60-90 triangle to see how the sides change while staying in the 1 : 2 : 3 ratio.
sin60∘=23cos60∘=21tan60∘=13
Basic Concepts
remaining today
remaining today