Tag Archives: cosine

Right Triangle Trigonometry and SOHCAHTOA

Sohcahtoa isn't actually an Egyptian god, but if it helps to remember him that way, you'll have an easier time recalling right angle trig relationships.

Sohcahtoa isn’t actually an Egyptian god, but if it helps to remember him that way, you’ll have an easier time recalling right angle trig relationships.

Right triangles are extremely common in science homework. Even though they are common, they can be confusing to new students. That is why we have the Egyptian god SOHCAHTOA.

SOHCAHTOA is a handy mnemonic trigonometry students learn to remember which sides of a triangle are used for the three main trig functions: sine, cosine, and tangent.

These functions are defined by the ratios of various lengths of the sides of a right triangle. Let’s look at this right triangle.

Right TriangleThis triangle is made up of three sides of lengths a, b and c. Note the angle marked θ. This angle is formed by the intersection of b and c. The hypotenuse is always the longest of the three sides and opposite of the right angle. The side b is ‘adjacent’ to the angle, so this side is known as the adjacent side. It follows the side ‘opposite’ of the angle is known as the opposite side. Now that we have all our sides labelled, we can use SOHCAHTOA.

SOHCAHTOA

S – Sine
O – Opposite
H – Hypotenuse

C – Cosine
A – Adjacent
H – Hypotenuse

T – Tangent
O – Opposite
A – Adjacent

SOH = sin θ = opposite over hypotenuse = ac
CAH = cos θ = adjacent over hypotenuse = bc
TOA = tan θ = opposite over adjacent = ab

Easy to remember. Now let’s see how easy it is to apply.

Example Problem

Consider this triangle.

trig example for SOHCAHTOAThe hypotenuse has a length of 10 and one angle of the triangle is 40º. Find the lengths of the other two sides.

Let’s start with the side with length a. This side is opposite the angle and we know the length of the hypotenuse. The part of SOHCAHTOA with both hypotenuse and opposite is SOH or sine.

sin 40º = opposite / hypotenuse
sin 40º = a / 10

solve for a by multiplying both sides by 10.

10 sin 40º = a

Punch 40 into your calculator and hit the sin key to find the sine of 40º.

sin 40º = 0.643

a = 10 sin 40º
a = 10 (0.643)
a = 6.43

Now let’s do side b. This side is adjacent to the angle, so we should use CAH or cosine.

cos 40º = adjacent / hypotenuse
cos 40º = b / 10

solve for b

b = 10 cos 40º

Enter 40 and hit the cos button on your calculator to find:

cos 40º = 0.766

b = 10 cos 40º
b = 10 (0.766)
b = 7.66

The sides of our triangle are 6.43 and 7.66. We can use the Pythagorean equation to check our answer.

a2 + b2 = c2
(6.43)2 + (7.66)2 = c2
41.35 + 58.68 = c2
100.03 = c2
10.00 = c

10 is the length of the triangle’s hypotenuse and matches our calculation above.

As you can see, our friend SOHCAHTOA can help us calculate the angles and lengths of the sides of right triangles with very little information. Make him your friend too.

 

Downloadable Trig Table PDF

This table contains values for sine, cosine and tangent for angles between 0 and 90º. All values are rounded to four decimal places.

Trig TableClick the image for the full-sized image or download the PDF version.

The downloadable trig table PDF is optimized to fit on a single 8½ x 11″ sheet of paper.

Right Triangles – Trigonometry Basics

The right triangle is a special type of triangle that forms the basis for nearly all trigonometry functions.

As you know, a triangle is a closed shape consisting of three sides. Every two of these sides form an angle between them for a total of three interior angles. These three angles together add up to 180º. If one of these angles is 90º, also known as a right angle, the triangle is known as a right triangle.

Right TriangleThis is a right triangle. The three angles A, B and C are marked and each side’s length is denoted by a, b, and c. The side length letters are the sides opposite of the angle with the corresponding capital letter. The side opposite the right angle C is known as the hypotenuse of the right triangle.

The length of the hypotenuse can be calculated using the Pythagorean Theorem

a2 + b2 = c2

Each triangle has six parts, 3 sides and 3 angles. If you know two of these values, you can calculate any of the other four using the six trigonometric ratios. These ratios are sine, cosine, tangent, cosecant, secant, and cotangent. Using the variables in the right triangle, the formulas for these ratios for angle A are:

Sine formula

cosine formula

tangent formula

cosecant formula

secant formula

tangent formula

The common method to remember these formulas is the mnemonic SOH CAH TOA.

S – Sine
O – Opposite
H – Hypotenuse

C – Cosine
A – Adjacent
H – Hypotenuse

T – Tangent
O – Opposite
A – Adjacent

The last three are reciprocals of the first three. The tricky part is to remember cosecant goes with sine, not cosine as the name may suggest.

Memorize and learn these relationships of a right triangle. You will see these formulas again as you further your studies in science and engineering.