Sin, cos, and tan are three functions that each provide a relationship between two sides and one angle of a right-angled triangle – that is, a triangle with a 90° angle.
If we have some right angled triangle, we can mark one one the two acute angles with an X, and label the sides in the following way:
The side across from the right angle is the Hypotenuse. This is also the longest side. The side next to the angle X that is not the Hypotenuse is the Adjacent side. “Adjacent” means “next to.” The side across from the X is the Opposite side.
We then have the following relations:
sin X = Opposite / Hypotenuse
cos X = Adjacent / Hypotenuse
tan X = Opposite / Adjacent
One way to remember these relations is with the acronym SOH CAH TOA. These equations are useful for finding side lengths and the values of angles.
Finding Sides
In the above example, if:
Adjacent = 10
X = 45°
Then we can say tan 45° = Opposite / 10. Solving the equation for the Opposite side, we get:
1 = Opposite / 10
10 = Opposite
And we have that the length of the Opposite side is 10 units.
We could also say that cos 45° = 10/ Hypotenuse. Solving this equation for the Hypotenuse, we get:
0.7071 = 10 / Hypotenuse
Hypotenuse * 0.7071 = 10
Hypotenuse = 10 / 0.7071
Hypotenuse = 14.14
And we have that the length of the Hypotenuse is 14.14 units.
Finding Angles
We can also use the sin, cos, and tan relationships to find angle values if we know the lengths of two or more sides.
In the above example, if:
Opposite = 15
Hypotenuse = 30
Then we can say sin X = 15 / 30. Solving for X, we get:
sin X = 0.5
X = 30°
And we have that X is an angle of 30°.
Hopefully this guide helped you to gain a better understanding of how to use sin, cos, and tan!
Interested in math tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in New Haven, CT: visit: Tutoring in Alpharetta, GA.
