Enrol to start learning
Reading is open to everyone. Enrolling is free, and it is what unlocks the audio lessons, practice tests and progress tracking.
7.2. Example 2
Interactive Audio Lesson
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free account
Today we're going to discuss trigonometric ratios for right-angled triangles. Can anyone tell me what the three primary ratios are?
Is it sine, cosine, and tangent?
Exactly! So, remember the acronym ‘SOH-CAH-TOA’ to recall the relationships: Sin is Opposite over Hypotenuse, Cos is Adjacent over Hypotenuse, and Tan is Opposite over Adjacent.
What does 'adjacent' mean in this context?
Great question! The adjacent side is the one that is next to the angle we're looking at, excluding the hypotenuse. Let's put this into practice with an example.
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free account
If I have a right triangle where the hypotenuse measures 8 cm and the angle 𝜃 is 30°, how would we find the opposite side?
We can use the sine function!
Right! So, we apply the formula: Opposite = Hypotenuse × sin(𝜃). What do we get?
Opposite = 8 × sin(30°), which is 4 cm.
Excellent! That’s how we determine the length of the sides using our trigonometric ratios.
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free accountNow, what if we know the opposite side is 5 cm and the adjacent side is 12 cm? How can we find angle 𝜃?
We can use the tangent function's inverse!
Yes! We would calculate this as follows: 𝜃 = tan^(-1)(Opposite/Adjacent). Can someone calculate that?
So, 𝜃 = tan^(-1)(5/12). This gives us approximately 22.6°.
Correct! Understanding these functions not only helps in geometry but also in various applications in physics and engineering.
Overview
Short Summary
This section focuses on using trigonometric ratios to solve problems involving right-angled triangles.
Medium Summary
In this section, we apply trigonometric ratios to solve for unknown sides and angles in right-angled triangles, emphasizing the importance of understanding relationships and calculations between these components.
Detailed Summary
Example 2
In this section, we explore how to utilize trigonometric ratios to solve problems involving right-angled triangles. Trigonometric ratios, specifically sine, cosine, and tangent, enable us to find unknown lengths and angles when certain parameters are provided. These relationships are essential in practical applications across various fields. Understanding how to correctly apply these ratios, along with the use of inverse trigonometric functions, empowers us to navigate different geometrical challenges effectively.
Audio Book
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountSolution: Hypotenuse = \frac{7 cm}{\cos 45°} = 7 × \sqrt{2} ≈ 9.9 cm.
Detailed Explanation
To compute the hypotenuse, we first calculate . The cosine of 45 degrees equals or approximately 0.707. Plugging this value into the rearranged formula gives us:
This simplifies to:
Approximating as about 1.414, we find that the hypotenuse is approximately 9.9 cm.
Examples & Analogies
Continuing with the hill analogy, once you calculate how tall you would need to climb, you realize that the distance you need to cover to get to the peak (the hypotenuse) ends up being roughly 9.9 cm. This gives you a clear understanding of both the distance across the ground and the total distance you'd need to travel to reach the top of the hill.
--
Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Trigonometric Ratios: The relationships between the angles and sides of triangles, fundamental for solving triangle-related problems.
Reciprocal Ratios: Include cosecant, secant, and cotangent, which are ratios derived from the primary trigonometric functions.
Inverse Functions: Used to compute angles when two side lengths are known in a right triangle.
Examples
Memory Aids
Interactive tools to help you remember key concepts
Stories
Flash Cards
Glossary
Trigonometric Ratios
Ratios of the lengths of the sides of a right triangle; includes sine, cosine, and tangent.
Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Adjacent Side
The side next to the angle θ in a right-angled triangle, excluding the hypotenuse.
Opposite Side
The side opposite to the angle θ in a right-angled triangle.
Inverse Trigonometric Functions
Functions that allow for the calculation of angles from given side lengths.