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.
Today we are going to learn about basic trigonometric ratios. These ratios are crucial for understanding properties of right-angled triangles. Let’s start with sine. Who can tell me what sine is?
Isn't sine the ratio of the opposite side to the hypotenuse?
Exactly! We can remember this with the acronym SOH, which stands for 'Sine = Opposite over Hypotenuse'. Now, what about cosine?
Cosine is the adjacent side divided by the hypotenuse, right? That’s CAH.
Great job! CAH stands for 'Cosine = Adjacent over Hypotenuse'. Let’s move on to tangent. Can anyone tell me the definition of tangent?
Tangent is the opposite side divided by the adjacent side, which we can remember with TOA!
Fantastic! TOA stands for 'Tangent = Opposite over Adjacent'. Remembering these acronyms will help you recall the definitions easily. Does anyone have questions about these ratios?
Let’s discuss where we can apply these trigonometric ratios. Who can give me an example of where we might use sine?
We could use sine to calculate the height of a building if we know the angle of elevation from a certain distance!
Exactly! The sine ratio helps us find heights and distances in many practical scenarios. What about cosine?
Cosine can help in navigation, right? Like determining how far away something is when you know the angle and one side.
Yes, precisely! Cosine has many applications in navigation and physics. And tangent?
Tangent can be used to find slopes, like in roads or hills.
Excellent! Recognizing these applications makes it easier to see the relevance of trigonometric ratios in everyday life.
Now, let’s practice applying these ratios. If we have a right triangle with one angle θ=30° and we know the opposite side is 4 units, how can we find the hypotenuse?
We can use sine! Since , we can set up the equation: .
Correct! Now, what’s the hypotenuse?
The hypotenuse would be 8 units!
Well done! Practicing these calculations will help you become very skilled in applying trigonometric ratios. Let’s try one more problem involving cosine.
Example 1: In a right triangle where θ = 30°, if the opposite side is 3 units, then the hypotenuse can be found using sin θ = opposite/hypotenuse, giving hypotenuse = 3/sin(30°) = 6 units.
Example 2: For a right triangle with θ = 45°, if the adjacent side is 5 units, the hypotenuse becomes hypotenuse = 5/cos(45°) = 5√2 units.
In right triangles, use SOH CAH TOA, to find ratios, you'll go far!
Imagine climbing a mountain (sine), standing on the edge of a cliff (cosine), and finding a path (tangent). Each step up shows the ratio of sides!
SOH CAH TOA helps us remember 'Sine = Opposite over Hypotenuse', 'Cosine = Adjacent over Hypotenuse', and 'Tangent = Opposite over Adjacent'.
SOH CAH TOA for Sine, Cosine, and Tangent.
Term
What is Sine?
Click to flip
Term
What is Cosine?
Click to flip
Term
What is Tangent?
Click to flip
A trigonometric ratio defined as the ratio of the opposite side to the hypotenuse in a right triangle.
A trigonometric ratio defined as the ratio of the adjacent side to the hypotenuse in a right triangle.
A trigonometric ratio defined as the ratio of the opposite side to the adjacent side in a right triangle.
This section covers the fundamental trigonometric ratios related to right-angled triangles, including sine, cosine, and tangent.
In this section, we explore the basic trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios connect the angles and sides of right-angled triangles and are essential for various applications in geometry and trigonometry.
In this section, we focus on the basic trigonometric ratios that are essential for understanding trigonometry, particularly in the context of right-angled triangles. For a given angle θ in a right triangle:
Sine (sin) is defined as the ratio of the length of the opposite side to the hypotenuse:
Cosine (cos) is defined as the ratio of the length of the adjacent side to the hypotenuse:
Tangent (tan) is defined as the ratio of the length of the opposite side to the adjacent side:
These ratios are fundamental in various applications of mathematics, physics, and engineering, helping to solve problems involving angles and distances.