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### Course: Trigonometry>Unit 2

Lesson 1: Unit circle introduction

# The trig functions & right triangle trig ratios

Sal shows how, for acute angles, the two different definitions of the trigonometric values (SOH CAH TOA and the unit circle definition) result in the same values. Created by Sal Khan.

## Want to join the conversation?

• Hard to pay attention to, all I see is letters and those don't add or divide. they are not numbers. letters everywhere and confused
• Think of them as numbers with no identity, they are called variables.
• The video doesn't really help with the practice exercises, i've seen similar complaints below that are from a year ago and the issue has not been fixed since then... i'm jumping to the next section for now but please fix the issue, so we can come back and finish this section.

I've got two exercises out of five right in a row and as soon as i think to have got it, something blows up, it's so frustrating...
• Still kinda confused on the fractions on the unit circle, hard to remember.
• is to illustrate the fact their either the unit circle or a plain triangle can be used to find the same trig ratios
• In the unit circle, can we imagine the hypotenuse in any rt angle triangle is a straight line on a cartesian plane and so it's equation is y=mx+c, and, since c = zero, m=y/x - the equation for tan(theta). So tan(theta) is the gradient of the hypotenuse?
• Yes, your reasoning is perfectly right. Tangent is defined by the Y ordenade of the point in which the line defined by the angle crosses the x=1 line.
• What does Sal mean by the Unit Circle is an extension of SOH-CAH-TOA? I know what they mean, but how is it an extension?
• In SOH, CAH, TOA, definition of trig functions, we defined the sin,cos etc .
as the ratio of the sides of a triangle. Also, we were only able to find the value of trig functions of angles upto 90 degrees.
But in unit circle definition, the trigonometric functions sine and cosine are defined in terms of the coordinates of points lying on the unit circle x^2 + y^2=1. And unlike the previous definition(SOH, CA...), we are able to find sin,cos.... of all possible angles.
That's how the Unit Circle definition is an extension of SOH,CA.... definition.
• Not much explanation of what the ratios are or how they are used. Just a lot of quick talk. The exercise questions do not follow the lesson and have that bland text-book feel to them when you use hints. Khan Academy has definitely moved away from the simple approach that it was founded on to a more public school diploma chugging factory approach...
• In a physical sense, how can sin be applied to anything 90 or above, or 0 or below? Same thing for Cos and Tan?
• trigonometric functions like sine, cosine, and tangent have broad applicability in physical scenarios that involve angles both greater than 90 degrees and less than 0 degrees. These functions help describe various aspects of oscillatory motion, periodic phenomena, slopes, and ratios, making them essential tools in many scientific and engineering disciplines. The choice of angle measurement, whether in degrees or radians, depends on the specific context and conventions used in a particular field of study.

When dealing with angles greater than 90 degrees, the sine function is used to model oscillatory phenomena that extend beyond the initial position. For example, in mechanical engineering, you might use sine functions to describe the motion of a pendulum that swings past its vertical equilibrium point.

Negative angles can represent situations where you are measuring an angle in the opposite direction. For example, if you have a rotating object that starts at a certain position and moves counterclockwise, you can use negative angles to measure its position relative to the starting point. The sine function can still be used to calculate the vertical displacement from this position.

The cosine function is used to describe phenomena that involve periodic motion or oscillations. For instance, in electrical engineering, you might use cosine functions to model alternating current (AC) voltage or current, which oscillates above and below zero.

like with sine, negative angles can be used to measure positions in the opposite direction. The cosine function can still be employed to calculate horizontal displacements or projections in these cases.

Tangent is used to describe the steepness or slope of a line, including lines that are nearly vertical. In physics, this can be applied when studying inclined planes, projectile motion, or other scenarios involving angles greater than 90 degrees.

Negative angles can be used to describe directions opposite to the initial reference direction. Tangent can be applied to calculate the slope or ratio of vertical to horizontal displacement for angles in these directions.
• How can I solve for a trigonometric function without using my calculator? What does mathematicians do to solve a sine or inverse sine function in middle ages?
• According to Wikipedia, "Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering."
• Still kinda confused on the fractions on the unit circle, hard to remember.
• Which fractions are you talking aobut, the trig fractions as shown by SOH CAH TOA, fractions created by points at various angles such as 30, 60, 90, etc. or the fractions in radians related to these degree angles? There are all sorts of fractions related to the unit circle. I also notice that this is not even your own question, it is directly copied from the second most popular post, come up with origin question because directly copying and acting like it is your own is plagiarism.
• What happens if there is no unit circle? Do you just have to use a calculator or is there another way?
• While the unit circle is a useful visual aid for understanding and calculating trigonometric functions, it's not the only way to work with trigonometry. Trigonometric functions can be defined and calculated independently of the unit circle, and you can use other methods, including calculators and tables, to determine their values.

Here are some alternative methods for calculating trigonometric functions when you don't have access to a unit circle:

Calculator: Modern scientific and graphing calculators have built-in trigonometric functions that can directly compute the sine, cosine, tangent, and other trigonometric values for any given angle. You can enter an angle in degrees or radians, and the calculator will provide the corresponding trigonometric value.

Trigonometric Identities: Trigonometric identities, such as the Pythagorean identities and sum/difference identities, can be used to express trigonometric functions in terms of other trigonometric functions. These identities can be helpful for simplifying trigonometric expressions and solving trigonometric equations.

Trigonometric Tables: Before the advent of calculators, trigonometric tables were widely used to look up values of trigonometric functions for specific angles. These tables provide precomputed values for common angles, and you can interpolate between values for angles that are not listed.

Trigonometric Formulas: Trigonometric functions can be expressed in terms of complex numbers using Euler's formula

This approach is particularly useful in advanced mathematics and engineering, as it allows you to work with complex exponentials to solve various problems.

Numerical Methods: In some cases, when dealing with non-standard angles or complex problems, numerical methods like approximation algorithms or calculus techniques may be employed to calculate trigonometric values with precision.

While the unit circle provides an intuitive geometric interpretation of trigonometric functions, it's not necessary for performing trigonometric calculations. In practice, calculators and trigonometric tables are the most common tools for obtaining trigonometric values quickly and accurately.