Values of Trigonometric Ratios for Standard Angles
This page should serve as a useful reference point for all trigonometric values, including the values of standard angles and their measures in both degrees and radians. You can refer to this page to create flashcards, graphs, and other useful study tools that may help you memorize these key concepts. Once you have these values and ratios memorized, it should be easy to solve a wide range of geometric ratios with a few swipes of your pencil!
A review of trigonometric ratios
The most important trigonometric ratios are cosine, sine, and tangent. Let's review these three key concepts:
- Sine: This trigonometric ratio is the length of the leg opposite the angle divided by the length of the hypotenuse. In other words, sine = opposite/hypotenuse.
- Cosine: This trigonometric ratio is the length of the leg adjacent to the angle divided by the length of the hypotenuse. In other words, cosine = adjacent/hypotenuse
- Tangent: This trigonometric ratio is the length of the leg opposite to the angle divided by the leg adjacent to the angle. In other words, tangent = opposite/adjacent
Let's quickly review a few important definitions:
- Adjacent: The side adjacent to the angle is neither the hypotenuse nor the opposite side. It is the only one remaining, and it is located on one side of the angle.
- Opposite: The opposite side is exactly what it sounds like: This side is directly facing the angle, across the triangle in a diagonal manner.
- Hypotenuse: This fancy-sounding word is quite simple to understand: The hypotenuse is the side opposite the right angle.
Understanding the trigonometric values of standard angles
Standard angles are the most common angles. We see these angles all the time in tests and quizzes, so it makes sense to memorize their values if possible. When we know these values off the top of our heads, we can solve many triangles (and a wide range of other shapes) with just a few easy steps.
- The 0-degree angle: A zero-degree angle does not really exist, but it may still be useful to memorize its values. It has 0 radians, a sine value of zero, a cosine value of 1, and a tangent value of 0.
- The 30-degree angle: Memorizing the values of a 30-degree angle may be helpful when dealing with a 30-60-90 triangle. This angle has $\frac{\pi}{6}$ radians, a sine value of $\frac{1}{2}$ , a cosine value of $\frac{\sqrt{3}}{2}$ , and a tangent value of $\frac{\sqrt{3}}{3}$ .
- The 45-degree angle: 45-degree angles are very common, and memorizing their values can be especially useful when solving 45-45-90 triangles. These are also known as "isosceles right triangles." A 45-degree angle has $\frac{\pi}{4}$ radians, a sine value of $\frac{\sqrt{2}}{2}$ , a cosine value of $\frac{\sqrt{2}}{2}$ , and a tangent value of 1. This might be an easy angle to memorize because its sine and cosine values are equal.
- The 60-degree angle: Memorizing the values of a 60-degree angle make dealing with 30-60-90 triangles easier. These are special triangles with angles in the ratio 1:2:3. The 60-degree angle has $\frac{\pi}{3}$ radians, a sine value of $\frac{\sqrt{3}}{2}$ , a cosine value of $\frac{1}{2}$ , and a tangent value of $\sqrt{3}$ .
- The 90-degree angle: The 90-degree angle is extremely common, and it is very useful to memorize its values -- perhaps more so than any other angle on this list. Because of this, it's a good idea to start by memorizing the values of the 90-degree angle before steadily memorizing others. 90-degree angles are present in all right triangles. The 90-degree angle has $\frac{\pi}{2}$ radians, a sine value of 1, and a cosine value of 0. Its tangent is undefined. Why? Because you can't divide by zero.
- The 180-degree angle: A 180-degree angle has $\pi $ radians, a sine value of zero, a cosine value of -1, and a tangent value of zero. Note that the tangent and sine values are the same -- making this another easy angle to memorize.
- The 270-degree angle: A 270-degree angle has $\frac{3\pi}{2}$ radians, a sine value of -1, a cosine value of 0, and an undefined tangent value.
- The 360-degree angle: As we all know, a 360-degree angle is the same as a complete circle. A 360-degree angle has $2\pi $ radians, a sine value of zero, a cosine value of 1, and a tangent value of zero.
The unit circle is essential in trigonometry for understanding angles, connecting radians and degrees, and simplifying common values for trigonometric functions.
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