If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Trigonometry>Unit 1

Lesson 5: Sine and cosine of complementary angles

# 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 ${30}^{\circ }$ angle and a ${60}^{\circ }$ angle.
45-45-90 triangles
A 45-45-90 triangle is a right triangle with two ${45}^{\circ }$ angles.

## The trigonometric ratios of ${30}^{\circ }$‍

We are now ready to evaluate the trig functions of these special angles. Let's start with ${30}^{\circ }$.
Study the worked example below to see how this is done.

### What is $\mathrm{sin}\left({30}^{\circ }\right)$‍ ?

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 $1x$ so that it is clear that $\frac{x}{2x}=\frac{1x}{2x}=\frac{1}{2}$.
Now let's use this method to find $\mathrm{cos}\left({30}^{\circ }\right)$ and $\mathrm{tan}\left({30}^{\circ }\right)$.

## The trigonometric ratios of ${45}^{\circ }$‍

Let's try this process again with ${45}^{\circ }$. Here we can start by drawing and labeling the sides of a 45-45-90 triangle.

## The trigonometric ratios of 60${}^{\circ }$‍

The process of deriving the trigonometric ratios for the special angles ${30}^{\circ }$, ${45}^{\circ }$, and ${60}^{\circ }$ is the same.
While we have not yet explicitly shown how to find the trigonometric ratios of ${60}^{\circ }$, we have all of the information we need!

## A summary

We have calculated the trig ratios for ${30}^{\circ }$, ${45}^{\circ }$, and ${60}^{\circ }$. The table below summarizes our results.
$\mathrm{cos}\left(\theta \right)$$\mathrm{sin}\left(\theta \right)$$\mathrm{tan}\left(\theta \right)$
$\theta ={30}^{\circ }$$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$\frac{\sqrt{3}}{3}=\frac{1}{\sqrt{3}}$
$\theta ={45}^{\circ }$$\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$$\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$$1$
$\theta ={60}^{\circ }$$\frac{1}{2}$$\frac{\sqrt{3}}{2}$$\sqrt{3}$
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?
• No. The trig functions are not linear; therefore, you cannot use ratios like that.
• why does sal specify that tangent of 30 degrees is √3/3 ? isn't it 1/√3
• 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
• what are the sin, cos and tan of 90 degrees?
• 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.
• 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?
• Sin is opposite/hypotenuse. So you just need a right triangle with that angle then do o/h.
(1 vote)
• I don't get how side opposite of the 30-degree angle is x/2. can someone help me understand this problem.
• 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
• 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?
• 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.
• 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.
• Is there a specific value of sin, cos or tan or are they just like a placeholder with different values each time like x,y,z?
Basically speaking, does sin have a specific value when it is 100%?
• sin(x), cos(x) and tan(x) are functions, so their value depends on what value of x you input. Sure, you can use different variables (like sin(z) or tan(m)) but "sin", "cos" and "tan" by themselves have no meaning.
• 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?
• 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
• 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!