Trigonometry: Sine, Cosine, Tangent
The ratios that unlock right triangles
The ratios that unlock right triangles
Before we dive into trigonometry, we need to make sure we know the parts of a right triangle. A right triangle has three sides and three angles, and one angle is always exactly 90 degrees — that is the right angle. The side opposite the right angle is the longest side, and it has a special name: the hypotenuse. The other two sides are called the legs. The hypotenuse is always across from the right angle, and the two legs form the right angle between them.
In trigonometry, we label the triangle relative to one of the non-right angles. We pick an angle (let's call it angle A), and then we name the sides based on where they are in relation to that angle. The side across from angle A is called the opposite side. The side next to angle A that is not the hypotenuse is called the adjacent side. The remaining side, across from angle A's complement, is the hypotenuse. This labeling changes if we pick a different angle, but the hypotenuse always stays the same.
Understanding this labeling is crucial because the three trigonometric ratios — sine, cosine, and tangent — are all defined in terms of opposite, adjacent, and hypotenuse relative to a chosen angle. Once you get comfortable with this labeling, the rest of trigonometry starts to fall into place.
SOH CAH TOA is a mnemonic that helps you remember the three basic trigonometric ratios. SOH stands for Sine equals Opposite over Hypotenuse. CAH means Cosine equals Adjacent over Hypotenuse. TOA tells us Tangent equals Opposite over Adjacent. Each ratio compares two sides of a right triangle, and each tells us something useful about the angles inside the triangle.
Sine (sin) is the ratio of the opposite side to the hypotenuse. If you have a right triangle where the side opposite your chosen angle is 3 units long and the hypotenuse is 5 units long, then sin of that angle is 3 divided by 5, which is 0.6. Cosine (cos) is the ratio of the adjacent side to the hypotenuse. If the adjacent side is 4 units and the hypotenuse is 5 units, then cos of that angle is 4 divided by 5, which is 0.8. Tangent (tan) is the ratio of the opposite side to the adjacent side. In this example, that would be 3 divided by 4, which is 0.75.
These three ratios — sin, cos, and tan — are the foundation of all trigonometry. They relate the angles of a right triangle to the lengths of its sides. Once you know any two sides of a right triangle, you can find the ratios for its angles, and from those ratios, you can find the angles themselves.
Calculators are essential tools for trigonometry beyond simple textbook triangles. On most scientific calculators, you will find buttons labeled SIN, COS, and TAN. To use them, you enter an angle measure and then press the button, or vice versa depending on the calculator. Most calculators work in degrees, so make sure your calculator is set to DEG mode, not RAD or GRAD, unless the problem specifies otherwise.
For example, if you want to find the sine of 30 degrees, you would press 30 and then the SIN button. The calculator will give you 0.5, which is correct. To go the other way — to find an angle when you know one of its trig ratios — you use the inverse functions, usually labeled SIN inverse (or SIN with a negative one superscript: sin minus one), COS inverse, and TAN inverse. If you know that the sine of an angle is 0.5, pressing SIN inverse of 0.5 will give you 30 degrees.
Practicing with your calculator is important. Make sure you understand how to switch between normal and inverse modes. Try working through several examples until using these functions feels automatic. A common mistake is forgetting to check whether the calculator is in degree or radian mode, which can produce wildly wrong answers.
One of the most practical uses of trigonometry is finding the length of a side of a right triangle when you know one side and one angle (other than the right angle). This comes up constantly in real-world situations like surveying, construction, and navigation. Here is the step-by-step approach.
Suppose you have a right triangle where one of the acute angles is 40 degrees, the side adjacent to that angle is 10 units long, and you need to find the length of the opposite side. You know the adjacent and need the opposite, so you should use the tangent ratio: tan(40 degrees) equals opposite divided by adjacent. Rearranging: opposite equals tan(40) times 10. Using a calculator, tan(40) is approximately 0.8391. Multiply by 10 to get approximately 8.39 units. If you needed the hypotenuse instead, you would use sine or cosine with the same angle.
Sometimes you know all three sides of a triangle and need to find the angles. In that case, you use the inverse trig functions. Say you have a right triangle with an opposite side of 6 and an adjacent side of 8. The tangent of your target angle is opposite divided by adjacent, which is 6 divided by 8, or 0.75. To find the angle, press TAN inverse of 0.75 on your calculator. The answer is approximately 36.87 degrees.
You can also use sine or cosine for this, depending on which sides you know. If you know the opposite and the hypotenuse, use sine inverse. If you know the adjacent and the hypotenuse, use cosine inverse. The key is to pick the ratio that matches the sides you have. With practice, you will develop an intuition for which ratio to use in different situations.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. In equation form: hypotenuse squared equals leg one squared plus leg two squared, or a squared plus b squared equals c squared. This theorem works alongside trigonometry — it lets you check your answers or find a third side when you know the other two.
Notice something interesting about the sine and cosine values we calculated earlier. For the triangle with opposite 3, adjacent 4, and hypotenuse 5, we had sin equals 3/5 (0.6) and cos equals 4/5 (0.8). If you square both and add them, you get (3/5) squared plus (4/5) squared equals 9/25 plus 16/25 equals 25/25 equals 1. This is the fundamental trigonometric identity: sin squared theta plus cos squared theta equals 1. This relationship is always true for any angle theta, and it is one of the most useful identities in all of trigonometry.
Trigonometry is not just a math topic you study and forget — it has countless real-world applications that affect our daily lives. Surveyors use trigonometry to measure distances and angles across large areas of land. When you see a construction crew measuring the slope of a hill, they are almost certainly using trig. Architects use trigonometry to calculate roof slopes, structural loads, and ceiling heights. Engineers use trig to design bridges, tunnels, and roads that involve slopes and curves.
Navigation is built on trigonometry. When you use a GPS, the system is solving trigonometric equations using signals from satellites to pinpoint your location. Astronomers use trigonometry to measure the distances to nearby stars using a technique called stellar parallax. Even video game designers use trigonometry to calculate how objects move on screen, especially in games where characters jump, arc through the air, or rotate around a center point.
Perhaps the most relatable example is measuring the height of something tall without climbing it. If you stand a known distance from a tree and measure the angle to the top, you can calculate the tree's height using trigonometry. Height equals the adjacent side times the tangent of the angle. This is exactly how surveyors measured mountains and buildings before drones and laser technology became available.