Lesson 7.1: Right-angled Triangle Trigonometry
Introduction
The objective of this lesson is to introduce you, students, to the fascinating world of trigonometry as it applies to right-angled triangles. Trigonometry is fundamental in various scientific fields, and it will help you understand relationships between angles and sides in triangles. By the end of this lesson, you will be able to:
- Understand and use the sine, cosine, and tangent ratios in right-angled triangles.
- Find unknown sides and angles in these triangles.
- Apply Pythagoras' theorem alongside trigonometric ratios.
- Choose the correct ratio based on specific problems.
To begin, let’s consider a right-angled triangle, which is defined as a triangle that has one angle measuring $90^\circ$ (a right angle). The sides of a right-angled triangle are referred to as the opposite side, adjacent side, and hypotenuse. The hypotenuse is the longest side, opposite the right angle.
Trigonometric Ratios
In a right-angled triangle, there are three primary trigonometric ratios that are essential for solving problems: sine, cosine, and tangent. Each ratio relates an angle to the lengths of two of the triangle's sides.
Sine Ratio
The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Mathematically, this is expressed as:
$$
$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
$$
where $θ$ is the angle of interest.
Example 1: Finding the Sine Ratio
Consider a right-angled triangle where one angle $θ$ is $30^\circ$. The length of the hypotenuse is $10$ units, and we want to find the length of the opposite side.
Using the sine ratio, we have:
$$
$\sin(30^\circ) = \frac{\text{opposite}}{10}$
$$
We know that $\sin(30^\circ) = \frac{1}{2}$, so we substitute:
$$
$\frac{1}{2} = \frac{\text{opposite}}{10}$
$$
To solve for the opposite side, multiply both sides by $10$:
$$
\text{opposite} = $10 \cdot$ $\frac{1}{2}$ = $5 \text{ units}$
$$
Cosine Ratio
The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. This is expressed as:
$$
$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
$$
Example 2: Finding the Cosine Ratio
Using the same triangle with angle $θ = 30^\circ$ and hypotenuse $10$ units, we want to find the length of the adjacent side.
Using the cosine ratio, we have:
$$
$\cos(30^\circ) = \frac{\text{adjacent}}{10}$
$$
Knowing that $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, we substitute:
$$
$\frac{\sqrt{3}}{2} = \frac{\text{adjacent}}{10}$
$$
Multiply both sides by $10$ to isolate the adjacent side:
$$
\text{adjacent} = $10 \cdot$ $\frac{\sqrt{3}}{2}$ = $5\sqrt{3}$ $\text{ units}$
$$
Tangent Ratio
The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. This is expressed as:
$$
$ an(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
$$
Example 3: Finding the Tangent Ratio
Using the lengths we found in the previous examples, where the opposite side is $5$ units and the adjacent side is $5\sqrt{3}$ units, we want to calculate the tangent of $30^\circ$.
Using the tangent ratio, we have:
$$
$ an(30^\circ) = \frac{5}{5\sqrt{3}}$
$$
This simplifies to:
$$
$ an(30^\circ) = \frac{1}{\sqrt{3}}$
$$
Applying Pythagoras' Theorem
In trigonometry, Pythagoras' theorem plays a crucial role and serves as a foundation for finding unknown sides. It states that in a right-angled triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a$ and $b$):
$$
$ c^2 = a^2 + b^2 $
$$
Example 4: Using Pythagoras' Theorem
Consider a right-angled triangle where the lengths of the two legs are $3$ units (adjacent) and $4$ units (opposite). Let's find the hypotenuse.
Applying Pythagoras' theorem:
$$
$ c^2 = 3^2 + 4^2 $
$$
Calculating the squares results in:
$$
c^2 = 9 + 16 = 25
$$
Taking the square root gives:
$$
c = $\sqrt{25}$ = $5 \text{ units}$
$$
Finding Unknown Sides and Angles
To solve for unknown sides or angles in a right-angled triangle, we can use the trigonometric ratios along with Pythagoras' theorem. The strategy involves selecting the correct ratio based on the known and unknown sides relative to the angle of interest.
Example 5: Finding an Unknown Angle
Suppose you have a right-angled triangle where the length of the opposite side is $8$ units and the length of the hypotenuse is $10$ units. To find the angle $θ$, we can use the sine ratio:
$$
$\sin(θ) = \frac{8}{10}$
$$
Simplifying gives:
$$
$\sin(θ) = 0.8 $
$$
To find the angle, we take the inverse sine:
$$
$θ = \sin^{-1}(0.8) \approx 53.13^\circ $
$$
Example 6: Choosing the Correct Ratio
Suppose we need to find an unknown side in a right-angled triangle, given an angle $30^\circ$ and the adjacent side of $6$ units. Here, we want to use the cosine ratio:
$$
$\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} $
$$
Knowing that $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, we can set up:
$$
$\frac{\sqrt{3}}{2} = \frac{6}{\text{hypotenuse}} $
$$
Cross-multiplying gives:
$$
$\sqrt{3} \cdot \text{hypotenuse} = 12 $
$$
Solving for the hypotenuse results in:
$$
\text{hypotenuse} = $\frac{12}{\sqrt{3}}$ = $4\sqrt{3}$ $\text{ units}$
$$
Conclusion
In this lesson, we explored the foundational aspects of trigonometry related to right-angled triangles. We learned to identify and apply the sine, cosine, and tangent ratios, alongside Pythagoras' theorem. You, students, should feel confident in finding unknown sides and angles in triangles and choosing the appropriate ratio to solve problems. Trigonometry is a powerful tool in mathematics, and understanding these concepts lays the groundwork for further exploration in the subject.
Study Notes
- Defining the sides of a right-angled triangle: opposite, adjacent, hypotenuse.
- Trigonometric ratios:
- Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
- Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
- Tangent: $ an(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
- Pythagoras' theorem: $c^2 = a^2 + b^2$.
- Use sine, cosine, and tangent ratios appropriately based on known and unknown sides.
- Angle finding using inverse trigonometric functions.