4 ## THE 30°-60°-90° TRIANGLETHERE ARE TWO special triangles in trigonometry. One is the 30°-60°-90° triangle. The other is the isosceles right triangle. They are special because, with simple geometry, we can know the ratios of their sides. Theorem. We will prove that below. (For the definition of measuring angles by "degrees," see Topic 12.) Note that the smallest side, 1, is opposite the smallest angle, 30°; while the largest side, 2, is opposite the largest angle, 90°. (Theorem 6). (For, 2 is larger than . Also, while 1 : : 2 correctly corresponds to the sides opposite 30°-60°-90°, many find the sequence 1 : 2 : easier to remember.) The cited theorems are from the Appendix, Some theorems of plane geometry. Here are examples of how we take advantage of knowing those ratios. First, we can evaluate the functions of 60° and 30°. Example 1. Evaluate cos 60°.
Since the cosine is the ratio of the adjacent side to the hypotenuse, you can see that cos 60° = ½. Example 2. Evaluate sin 30°.
You can see that directly in the figure above. Problem 1. Evaluate sin 60° and tan 60°. To see the answer, pass your mouse over the colored area. The sine is the ratio of the opposite side to the hypotenuse.
The tangent is ratio of the opposite side to the adjacent.
Problem 2. Evaluate cot 30° and cos 30°. The cotangent is the ratio of the adjacent side to the opposite.
= . Or, more simply, cot 30° = tan 60°. Problem 1. As for the cosine, it is the ratio of the adjacent side to the hypotenuse. Therefore,
Before we come to the next Example, here is how we relate the sides and angles of a triangle: If an angle is labeled capital A, then the side opposite will be labeled small
Example 3. Solve the right triangle ABC if angle A is 60°, and side
Now in every 30°-60°-90° triangle, the sides are in the ratio 1 : 2 : , as shown on the right. Whenever we know the ratios of the sides, we can solve the triangle by the method of similar figures. And so in triangle ABC, the side corresponding to 2 has been multiplied by 5. Therefore every side will be multiplied by 5. Side Alternatively, we could say that the side adjacent to 60° is always Whenever we know the ratio numbers, the student should use this method of similar figures to solve the triangle, and not the trigonometric Table. (In Topic 6, we will solve right triangles the ratios of whose sides we do not know.)
Problem 3. In the right triangle DFE, angle D is 30°, and side DF is 3 inches. How long are sides
The student should draw a similar triangle in the same orientation. Then see that the side corresponding to was multiplied by . Therefore, each side will be multiplied by . Side
Problem 4. In the right triangle PQR, angle P is 30°, and side
The side corresponding to 2 has been Problem 5. Solve the right triangle ABC if angle A is 60°, and the hypotenuse is 18.6 cm.
The side adjacent to 60° is always Example 4. ABC is an equilateral triangle whose height AD is 4 cm. Find the length of the side
The height of a triangle is the straight line drawn from the vertex at right angles to the base. Therefore, triangle ADB is a 30-60-90 triangle. For this problem, it will be convenient to form the proportion with fractional symbols:
That implies
On taking to be approximately 1
The side corresponding to was multiplied to become 4. How was it multiplied?
Compare the previous topic.
which, again, is approximately 4
Problem 6. Inspect the values of 30°, 60°, and 45° -- that is, -- and in each equation, decide which of those angles is the value of
Example 5. Solve this equation for angle
Now, the sine of what angle is ½?
Problem 7. Solve this equation for angle
Problem 8. Prove:
The area In an equilateral triangle each side is
Since sin 60° = ½ (Problem 1),
Problem 9. Prove:
The three radii divide the triangle into three congruent triangles (Side-side-side); hence each radius bisects each vertex into two 30° angles.
Therefore,
so that
Now, according to the previous problem, the area
Therefore,
That is what we wanted to prove.
Problem 10. Prove: Let ABC be an equilateral triangle, let AD, BF, CE be the angle bisectors of angles A, B, C respectively; then those angle bisectors meet at the point P such that AP is two thirds of AD. First, triangles BPD, APE are congruent.
For, since the triangle is equilateral and BF, AD are the angle bisectors, then angles PBD, PAE are equal and each
30°;
Therefore, BP = 2PD.
But AP = BP, because triangles APE, BPD are conguent, and those are the sides opposite the equal angles. The proof Here is the proof that in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : . It is based on the fact that a 30°-60°-90° triangle is Draw the equilateral triangle ABC. Then each of its equal angles is 60°. (Theorems 3 and 9) Draw the straight line AD bisecting the angle at A into two 30° angles. Now, since BD is equal to DC, then BD is half of BC. This implies that BD is also half of AB, because AB is equal to BC. That is, BD : AB = 1 : 2 From the Pythagorean theorem, we can find the third side AD:
Therefore in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : ; which is what we set out to prove. Corollary. The square drawn on the height of an equalateral triangle is three fourths of the square drawn on the side. Next Topic: The Isosceles Right Triangle Copyright © 2018 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |