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 explore right-angled triangles. Can anyone tell me what a right-angled triangle is?
Isn't it a triangle where one angle is 90 degrees?
That's correct! The side opposite the 90-degree angle is called the hypotenuse. Now, there are also two other sides we need to recognize.
What are those sides called, Teacher?
The other two sides are referred to as the 'opposite' side and the 'adjacent' side, depending on the angle we're discussing. Let's remember — the hypotenuse is the longest side!
How do these sides help us with trigonometry?
Great question! They help us define the trigonometric functions: sine, cosine, and tangent. Let's break those down now!
First up is sine! We define sine as the ratio of the opposite side to the hypotenuse. So if I write sin(θ) = opposite / hypotenuse, what do you think that describes?
It describes the sine of angle θ!
Exactly! Now, what about cosine?
Isn't that cos(θ) = adjacent / hypotenuse?
Spot on! And finally we have tangent, which is the ratio of opposite to adjacent. Can you write down tan(θ) = opposite / adjacent?
Why are there parts like adjacent and opposite in those definitions, Teacher?
Good question! Those terms depend on which angle of the triangle you are focusing on. It helps in accurately calculating the values.
Now that we've understood sine, cosine, and tangent, let's look at their reciprocals. Can anyone tell me what the reciprocal of sine is?
Is it cosecant?
Correct! We denote it as cosec(θ) = 1/sin(θ). What about for cosine?
That would be secant, right?
Yes! So, sec(θ) = 1/cos(θ). Finally, what's the reciprocal of tangent?
It's cotangent! So cot(θ) = 1/tan(θ).
Well done! Understanding these reciprocal functions will really help as you dive deeper into trigonometry.
Let's apply these definitions! If I have a right-angled triangle where the opposite side is 3 units and the hypotenuse is 5 units, how can I find sin(θ)?
We can use sin(θ) = opposite / hypotenuse, which would be 3/5.
Correct! And what would be the cosine in this case if the adjacent side is 4 units?
It would be cos(θ) = adjacent / hypotenuse, so 4/5!
Exactly! Now, let's summarize what we've learned today about right-angled triangles and their functions.
We learned about sine, cosine, tangent, and their reciprocal functions. They all relate to the sides of triangles!
Great recap! These relationships will help as we continue exploring trigonometry.
A triangle with one angle equal to 90 degrees, essential for defining trigonometric functions.
The longest side opposite the right angle in a right-angled triangle.
A function representing the ratio of the length of the opposite side to the hypotenuse.
A function representing the ratio of the length of the adjacent side to the hypotenuse.
A function representing the ratio of the length of the opposite side to the adjacent side.
Functions that are the inverse of the primary trigonometric functions, namely cosecant, secant, and cotangent.
Sine is opposite over the hype, it helps with triangles when it's ripe.
Imagine a right triangle named 'Riley' where Riley always remembers his opposite side is worthy of a party over the hypotenuse, which is the longest side. Together they show how to calculate sine!
For sine, think 'O/H' (Opposite over Hypotenuse), for cosine remember 'A/H' (Adjacent over Hypotenuse), and for tangent 'O/A' (Opposite over Adjacent).
For SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Term
What does sine represent in a right triangle?
Click to flip
Term
What is the tangent of angle θ?
Click to flip
Term
How do you find cosecant?
Click to flip
Term
What is the hypotenuse?
Click to flip
A triangle with one angle measuring 90 degrees.
The longest side of a right-angled triangle, opposite the right angle.
The side opposite to the angle in question.
The side next to the angle in question, excluding the hypotenuse.
The ratio of the opposite side to the hypotenuse of a right triangle.
The ratio of the adjacent side to the hypotenuse of a right triangle.
The ratio of the opposite side to the adjacent side of a right triangle.
The reciprocal of sine, defined as cosec(θ) = 1/sin(θ).
The reciprocal of cosine, defined as sec(θ) = 1/cos(θ).
The reciprocal of tangent, defined as cot(θ) = 1/tan(θ).
This section introduces the fundamental definitions related to right-angled triangles in trigonometry, including the key trigonometric functions derived from these definitions.
The section focuses on the definitions of three primary trigonometric functions: sine, cosine, and tangent, as they relate to right-angled triangles. It also highlights the reciprocal functions cosecant, secant, and cotangent, establishing a foundational understanding of these concepts within trigonometric functions.
In trigonometry, a right-angled triangle is defined as a triangle where one of the angles measures 90 degrees. The side opposite this angle is the hypotenuse, while the other two sides are referred to as the opposite and adjacent sides depending on the angle in consideration. The trigonometric functions defined in relation to this triangle are essential for evaluating angles and lengths in various applications.
The primary trigonometric ratios derived from a right-angled triangle are:
sin(θ) = opposite / hypotenuse.cos(θ) = adjacent / hypotenuse.tan(θ) = opposite / adjacent.Additionally, we define reciprocal functions for sine, cosine, and tangent:
cosec(θ) = 1/sin(θ).sec(θ) = 1/cos(θ).cot(θ) = 1/tan(θ).Understanding these definitions lays the groundwork for more advanced trigonometric calculations and identities that will be examined in subsequent sections.