Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Good morning everyone! Today, we're going to delve deeper into right-angled triangles. Can anyone tell me the names of the sides of a right-angled triangle?
The longest side is called the hypotenuse.
And there's the opposite side and the adjacent side!
Exactly! The hypotenuse is opposite the right angle. The opposite side is opposite the angle of interest, while the adjacent side is next to the angle of interest. To remember these names, think of the acronym 'OHA,' where O stands for Opposite, H for Hypotenuse, and A for Adjacent.
That's a good way to remember!
Let's then move on to how we can calculate these sides using trigonometric ratios!
Now that we know the sides, let's use trigonometric ratios to find unknown sides. If we know the hypotenuse and one angle, how do we find the opposite side?
We can use the sine function!
Correct! The formula is: Opposite = Hypotenuse × sin(θ). Let's say the hypotenuse is 10 cm and the angle θ is 30°. What would the opposite side be?
Using the sine formula: Opposite = 10 × sin(30°), which is 5 cm!
Excellent! Now, what if we wanted to find the adjacent side instead?
We would use cosine! Adjacent = Hypotenuse × cos(θ).
Awesome! Remember to practice these calculations, as they are fundamental in applying trigonometry in various scenarios.
Now, let's discuss how to find angles when we know two sides. What do you think we can use?
We can use inverse functions, like sin^(-1), cos^(-1), or tan^(-1)!
Absolutely! For example, if we have an opposite side of 5 cm and a hypotenuse of 10 cm, we can find angle θ by using θ = sin^(-1)(Opposite/Hypotenuse).
So that would be θ = sin^(-1)(5/10), which equals 30°!
Perfect! Remember to clarify which sides correspond to which ratios when applying these functions.
Let’s try putting all this into practice! If we have an angle of 45° and the adjacent side is 7 cm, how do we find the hypotenuse?
We use the cosine function! Hypotenuse = Adjacent/cos(θ).
Great! So, calculating Hypotenuse = 7 / cos(45°) gives us approximately 9.9 cm. Good job!
What about finding the side opposite?
In that case, we could use either the sine or tangent ratio. Using the sine function: Opposite = Hypotenuse × sin(θ).
Now I understand how to find both sides!
Exactly! Keep practicing, and soon finding unknown values in triangles will be a breeze.
Sine, Cosine, and Tangent relate side lengths to angles in right triangles.
Cosecant, Secant, and Cotangent are the reciprocals of the primary trigonometric ratios.
Used to find angles when the lengths of two sides are known.
Sin is opposite, on top it will lie, Cos is adjacent, as neat as a pie.
In a right triangle city, there lived a family. The Hypotenuse was the oldest, wise and tall. The Opposite, a dreamer, believed in the mystical heights, while the Adjacent, practical and clever, knew the straight paths well.
SOC: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
HAT: The Hypotenuse is the longest, the Adjacent is by the angle, and the Opposite is across.
Term
Sine Function
Click to flip
Term
Cosine Function
Click to flip
Term
Tangent Function
Click to flip
The longest side of a right-angled triangle opposite the right angle.
The side opposite to the angle of interest in a right-angled triangle.
The side next to the angle of interest in a right-angled triangle.
A trigonometric ratio defined as opposite side over hypotenuse.
A trigonometric ratio defined as adjacent side over hypotenuse.
A trigonometric ratio defined as opposite side over adjacent side.
Functions that can be used to find angles when two sides are known.
This section explains how to find the unknown sides and angles of a right-angled triangle using trigonometric ratios and inverse functions.
In this section, students learn the methods for calculating unknown sides and angles of right-angled triangles by utilizing trigonometric ratios such as sine, cosine, and tangent, as well as their inverses. Emphasis is placed on applying these concepts through real-world examples to enhance understanding.
In this section, we focus on how to find unknown sides and angles in right-angled triangles using trigonometric ratios. Given an angle 0 (other than the right angle) and one side of the triangle, we can use either sine or cosine to find the lengths of other sides.
Sine (sin): Ratio of the length of the opposite side to the hypotenuse.
Opposite Side: To find the length of the opposite side, we can use:
Opposite = Hypotenuse × sin(θ)Opposite = Adjacent × tan(θ)Adjacent Side: To find the adjacent side, we use:
Adjacent = Hypotenuse × cos(θ)Adjacent = Opposite / tan(θ)These formulas enable us to solve practical problems in various fields, such as physics and engineering, where right-angled triangles are common. Understanding these relationships can also deepen our grasp of trigonometry as a whole and its applications in real life.