Main content

## Trigonometry

### Course: Trigonometry > Unit 1

Lesson 5: Sine and cosine of complementary angles- Intro to the Pythagorean trig identity
- Sine & cosine of complementary angles
- Using complementary angles
- Relate ratios in right triangles
- Trig word problem: complementary angles
- Trig challenge problem: trig values & side ratios
- Trig ratios of special triangles

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Trig ratios of special triangles

Learn to find the sine, cosine, and tangent of 45-45-90 triangles and also 30-60-90 triangles.

Until now, we have used the calculator to evaluate the sine, cosine, and tangent of an angle. However, it is possible to evaluate the trig functions for certain angles without using a calculator.

This is because there are two

*special triangles*whose side ratios we know! These two triangles are the 45-45-90 triangle and the 30-60-90 triangle.## The special triangles

**30-60-90 triangles**

A 30-60-90 triangle is a right triangle with a 30, degrees degree angle and a 60, degrees degree angle.

**45-45-90 triangles**

A 45-45-90 triangle is a right triangle with two 45, degrees degree angles.

## The trigonometric ratios of 30, degrees

We are now ready to evaluate the trig functions of these special angles. Let's start with 30, degrees.

Study the worked example below to see how this is done.

### What is sine, left parenthesis, 30, degrees, right parenthesis?

Here's a worked example:

**Step 1:**Draw the special triangle that includes the angle of interest.

**Step 2:**Label the sides of the triangle according to the ratios of that special triangle.

**Step 3:**Use the definition of the trigonometric ratios to find the value of the indicated expression.

Note that you can think of x as 1, x so that it is clear that start fraction, x, divided by, 2, x, end fraction, equals, start fraction, 1, x, divided by, 2, x, end fraction, equals, start fraction, 1, divided by, 2, end fraction.

Now let's use this method to find cosine, left parenthesis, 30, degrees, right parenthesis and tangent, left parenthesis, 30, degrees, right parenthesis.

## The trigonometric ratios of 45, degrees

Let's try this process again with 45, degrees. Here we can start by drawing and labeling the sides of a 45-45-90 triangle.

## The trigonometric ratios of 60degrees

The process of deriving the trigonometric ratios for the special angles 30, degrees, 45, degrees, and 60, degrees is the same.

While we have not yet explicitly shown how to find the trigonometric ratios of 60, degrees, we have all of the information we need!

## A summary

We have calculated the trig ratios for 30, degrees, 45, degrees, and 60, degrees. The table below summarizes our results.

cosine, left parenthesis, theta, right parenthesis | sine, left parenthesis, theta, right parenthesis | tangent, left parenthesis, theta, right parenthesis | |
---|---|---|---|

theta, equals, 30, degrees | start color #1fab54, start fraction, square root of, 3, end square root, divided by, 2, end fraction, end color #1fab54 | start color #1fab54, start fraction, 1, divided by, 2, end fraction, end color #1fab54 | start color #1fab54, start fraction, square root of, 3, end square root, divided by, 3, end fraction, equals, start fraction, 1, divided by, square root of, 3, end square root, end fraction, end color #1fab54 |

theta, equals, 45, degrees | start color #aa87ff, start fraction, square root of, 2, end square root, divided by, 2, end fraction, equals, start fraction, 1, divided by, square root of, 2, end square root, end fraction, end color #aa87ff | start color #aa87ff, start fraction, square root of, 2, end square root, divided by, 2, end fraction, equals, start fraction, 1, divided by, square root of, 2, end square root, end fraction, end color #aa87ff | start color #aa87ff, 1, end color #aa87ff |

theta, equals, 60, degrees | start color #1fab54, start fraction, 1, divided by, 2, end fraction, end color #1fab54 | start color #1fab54, start fraction, square root of, 3, end square root, divided by, 2, end fraction, end color #1fab54 | start color #1fab54, square root of, 3, end square root, end color #1fab54 |

These values tend to occur often in advanced trigonometry problems. Because of this, it is helpful to know them.

Some people choose to memorize these values, but memorization is not necessary. In this article, you derived the values yourself, so hopefully you can re-derive them whenever you need them in the future.

## Want to join the conversation?

- Can these trigonometric rations of special triangles be used for finding the trigonometric ratios for other angles like 50 degrees? For example, if I did sin60 * 5/6, would that equal to sin50?(37 votes)
- No. The trig functions are not linear; therefore, you cannot use ratios like that.(91 votes)

- why does sal specify that tangent of 30 degrees is √3/3 ? isn't it 1/√3(26 votes)
- You are right but, we don't like to have the square roots on the bottom so we multiply both sides by the square root(9 votes)

- When I use a calculator and hit the "Sin" button (or another trig function button) and enter a given angle (ie. SIN(45)), what is the calculator doing to give me the opp/adj ratio?

Also, is there a way to find a ratio without using a calculator or a special triangle relationship? For example, how would you find what Sin(23.5) equals without a calculator?(7 votes)- Sin is opposite/hypotenuse. So you just need a right triangle with that angle then do o/h.(1 vote)

- what are the sin, cos and tan of 90 degrees?(3 votes)
- You can't really have a tangent of 90 degrees, at least when it comes to reference triangles, because that would indicate two 90 degree angles. As Sal mentioned in the videos talking about the Unit Circle, you can't have two 90 degree angles in a Triangle. You could theoretically have < or = 89.9999_` triangle, but theoretically two 90.00* = a line.(3 votes)

- I don't get how side opposite of the 30-degree angle is x/2. can someone help me understand this problem.(2 votes)
- When you divide an equilateral triangle into two, you have two 30-60-90 triangles of equal length. The opposite side of the 30 degree angle is the base. When you split the base of an equilateral triangle of side length x into 2, you get x/2. You can also try drawing it out.

Hope this helps(6 votes)

- In the second video, when Khan does the princibe root of sqrt2x^2 and c^2, how come the x goes outside and the two is left in?(2 votes)
- Its simple mathematics, think of this in this way:

2*x*x = c*c

we see 'x' and 'c' to be squared i.e.

2*x^2 = c^2

If you recall studying 'Real Numbers' or 'Indices', you would know that the root of 'x^2' is 'x' , the root of 'c^2' is 'c' and of '2' is '(root sign)2'.

So by simply finding root of both side:

2*x^2 = c^2, will be

(root)2*x = c

and so we write it as :

x*(root)2 = c,

and so here comes the concept that 2 is left behind, when truly it is not.(5 votes)

- I don't get how you're getting these answers, the math does not make sense. How is the Tan(30) not 1/√3 . I don't understand how you go from 1/√3 to √3/3. Same with Sin(45), how does he get √2/2? What am I missing here?(2 votes)
- you have to rationalize the denominator.

to rationalize 1/sqrt(3), you want to have no square root in the denominator, so you multiply numerator and denominator by sqrt(3): 1/sqrt(3)*sqrt(3)/sqrt(3)=sqrt(3)/(sqrt(3)*sqrt(3)).

sqrt(3)*sqrt(3)=3, so you get sqrt(3)/3.

1/sqrt(3) is equal to sqrt(3)/3, but they're just written differently because sqrt(3)/3 is considered simpler.

same goes to the sqrt(2)/2(4 votes)

- this is related to tan 60 and was a multiple-choice question in one of my papers;

the angle made by the line x/3 - y/2 = 1 with the positive direction of the x-axis is closest to:

then it gave some options, the correct answer being 33 degrees.

how do I get to that answer? I know that the gradient of the line is 2/3 and so the angle made by the line with the positive direction of the x-axis will be:

angle= tan-1 (inverse of tan) 2/3.

but how does this become close to 33 degrees?

I appreciate any help and thanks so much!(2 votes)- One of the options given is closest to the true measure of the angle.

That option is your answer.(4 votes)

- When looking for the cos(60º), I have the adj/hyp, which is X/2X. Is the reason that the answer is 1/2 because it is X over 2 times X and not X over 2X? (Because it makes more sense if it is X over 2 times X whereas if it was X over 2X then it would just be 1/X.)(3 votes)
- When I did math for cos60 I got 1 over square root of 2. The check said it was correct, but the table has square root of 2 over 2. Why the discrepancy? Same for tan30. I got 1 over square root of 3, but the chart says square root of 3 over 3.(1 vote)
- Because many tests require people to have a rational denominator, so we move the square root into the numerator.(2 votes)