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David Terr
Ph.D. Math, UC Berkeley

4.3. Right Triangle Trigonometry

In the previous section, we defined the six trigonometric functions in terms of the unit circle. For this reason, the trigonometric functions are also sometimes called circular functions.) However, there is another way to interpret trigonometric functions, namely in terms of right triangles. In fact, this is where the name trigonometry comes from. One should be careful, however, with this way of interpreting trigonometric functions, because these definitions only make sense for angles between 0 and 90 degrees (π/2 radians).

Consider the right triangle shown below. It has three sides, namely its two legs (the line segments forming the right angle) and the hypotenuse (the longest side). It also has three angles, a right angle (whose measure is π/2), the angle labeled θ in the diagram, and the third angle whose measure is π/2 - θ. (It is well -known that the sum of the three angles of every triangle is 180 degrees, or π radians.) The leg forming part of the angle θ is known as the adjacent,, and the other leg (opposite the angle θ) is known as the opposite. As shown in the diagram, the symbol A is used to denote the length of the adjacent, O is used for the length of the opposite, and H is used for the length of the hypotenuse.

In terms of the side of the right triangle, we then have the following definitions:

• (4.3.1) sin θ = O / H
• (4.3.2) cos θ = A / H
• (4.3.3) tan θ = O / A
• (4.3.4) cot θ = A / O
• (4.3.5) sec θ = H / A
• (4.3.6) csc θ = H / O

Armed with these definitions, we can now derive useful formulas for the values of the six trigonometric functions at the angles 30°, 45°, and 60° (π/6, π/4, and π/3 radians). First we consider a right triangle with angle 30° (π/6 radians) as shown below.

Note that this triangle is half of an equilateral triangle. Thus we see that the opposite side has half the length of the hypotenuse. In other words, we have

sin π/6 = O / H = 1/2.

Now the Pythagorean Theorem says that A2 + O2 = H2, implying that A = √H2-O2, whence

cos π/6 = A / H = √H2-O2 / H = √1-(O/H)2 = √1-(1/2)2 = √3/4 = √3/2.

Using Equations (4.2.3) - (4.2.6), it is now straightforward to compute the values of the other four trigonometric functions at θ = π/6. Specifically we have

tan π/6 = (sin π/6) / (cos π/6) = 1/√3 = √3/3

cot π/6 = 1 / (tan π/6) = √3

sec π/6 = 1 / cos π/6 = 2/√3 = 2√3/3

csc π/6 = 1 / sin π/6 = 2

The next special case to consider is a right triangle with an angle of 45 ° (π/4 radians) as shown below.

Note that the angle in the upper-right is also π/4, as it must be in order for the three angles to add up to π. This means that this triangle is an isosceles triangle as well as a right triangle. (All isosceles right triangles are similar to this one.) But we know that isosceles triangles, as well has having two equal angles, have two equal sides, which means O = A as indicated. Now by the Pythagorean Theorem, we have H2 = A2 + O2 = 2A2, whence H = A√2. Thus we see that

sin π/4 = cos π/4 = O / H = A / H = 1/√2 = √2/2.

We also have

tan π/4 = (sin π/4) / (cos π/4) = 1

cot π/4 = 1 / (tan π/4) = 1

sec π/4 = 1 / (cos π/4) = √2

csc π/4 = 1 / (sin π/4) = √2

The final case we will consider in this section is θ = π/3 (60 degrees) as shown below.

This case yields a right triangle similar to the one with θ = π/6 (30 degrees), the only difference being that the roles of the adjacent and opposite are now reversed. As a result, this triangle is also half of an equilateral triangle. Now it is clear, as shown in the diagram, that we have A = H / 2, whence

cos π/3 = A / H = 1/2

By the same arguments we used for the case of θ = π/6, we may derive the following additional identities. We leave the details as an exercise.

sin π/3 = O / H = √3/2

tan π/3 = (sin π/3) / (cos π/3) = √3

cot π/3 = 1 / (tan π/3) = 1/√3 = 3/3

sec π/3 = 1 / (cos π/3) = 2

csc π/3 = 1 / (sin π/3) = 2/√3 = 2√3/3

The following table sums up our knowledge of the values of all six trigonometric functions for the angles 0, π/6, π/4, π/3, and π/2. To get the values of the last four trigonometric functions for θ = 0 and θ = π/2, we used Equations (4.2.3) - (4.2.6). Note that some of the trigonometric functions are undefined for these values of θ; these entries are indicated with a dash. It is worth memorizing the values in this table at least for sin θ and cos θ, the others of which are easily derived from these.

θ (rad) θ (deg) sin θ cos θ tan θ cot θ sec θ csc θ
0
0
0
1
0
-
1
-
π/6
30
1/2
3/2
3/3
3
2√3/3
2
π/4
45
2/2
2/2
1
1
2
2
π/3
60
3/2
1/2
3
3/3
2
2√3/3
π/2
90
1
0
-
0
-
1

Table 4.3.1: Brief Table of Trigonometric Functions