Trigonometry Cheat Sheet Grade 10
๐ Introduction to Trigonometry
Trigonometry (from Greek trigลnon, "triangle" + metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. For grade 10, we primarily focus on right-angled triangles.
๐ History and Background
The earliest developments in trigonometry can be traced to ancient Egypt and Babylonia. Greek astronomers like Hipparchus (c. 190โ120 BC) are credited with creating the first trigonometric tables. The study of triangles and angles was crucial for astronomy and navigation.
๐ Key Principles: SOH CAH TOA
SOH CAH TOA is a mnemonic that helps remember the definitions of the trigonometric ratios:
- ๐ SOH: Sine = Opposite / Hypotenuse ($sin(\theta) = \frac{Opposite}{Hypotenuse}$)
- ๐ CAH: Cosine = Adjacent / Hypotenuse ($cos(\theta) = \frac{Adjacent}{Hypotenuse}$)
- ๐ TOA: Tangent = Opposite / Adjacent ($tan(\theta) = \frac{Opposite}{Adjacent}$)
๐งฎ Trigonometric Ratios
For a right-angled triangle with an angle $\theta$:
- ๐ Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
- ๐ Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
- โ๏ธ Tangent (tan): The ratio of the length of the side opposite the angle to the length of the adjacent side.
๐ Reciprocal Trigonometric Ratios
- cosecant$\csc(\theta) = \frac{1}{sin(\theta)}$
- secant$\sec(\theta) = \frac{1}{cos(\theta)}$
- cotangent$\cot(\theta) = \frac{1}{tan(\theta)}$
๐ Common Angles and Their Values
It's useful to memorize the trigonometric values for common angles:
| Angle ($\theta$) |
sin($\theta$) |
cos($\theta$) |
tan($\theta$) |
| 0ยฐ |
0 |
1 |
0 |
| 30ยฐ |
$\frac{1}{2}$ |
$\frac{\sqrt{3}}{2}$ |
$\frac{\sqrt{3}}{3}$ |
| 45ยฐ |
$\frac{\sqrt{2}}{2}$ |
$\frac{\sqrt{2}}{2}$ |
1 |
| 60ยฐ |
$\frac{\sqrt{3}}{2}$ |
$\frac{1}{2}$ |
$\sqrt{3}$ |
| 90ยฐ |
1 |
0 |
Undefined |
๐ก Practical Applications
- ๐ Navigation: Calculating distances and bearings.
- ๐๏ธ Engineering: Designing structures like bridges and buildings.
- ๐ญ Astronomy: Measuring the distances to stars and planets.
- ๐ฎ Game Development: Creating realistic movement and perspectives.
โ๏ธ Example Problem
A ladder leans against a wall, making an angle of 60ยฐ with the ground. If the foot of the ladder is 4 meters away from the wall, find the length of the ladder.
Solution:
Let the length of the ladder be $L$. We have $cos(60ยฐ) = \frac{4}{L}$.
Since $cos(60ยฐ) = \frac{1}{2}$, we get $\frac{1}{2} = \frac{4}{L}$.
Therefore, $L = 8$ meters.
๐ Practice Quiz
- If sin($\theta$) = 0.6, find cos($\theta$) and tan($\theta$).
- The angle of elevation of a ladder leaning against a wall is 60ยฐ and the foot of the ladder is 4.6 m away from the wall. What is the length of the ladder?
- Find the height of a tree if it casts a 10m shadow when the angle of elevation of the sun is 45ยฐ.
- A kite is flying at a height of 20 meters from the ground. The string attached to the kite makes an angle of 60ยฐ with the ground. Find the length of the string.
- A tower stands vertically on the ground. From a point on the ground which is 25m away from the foot of the tower, the angle of elevation of the top of the tower is 45ยฐ. What is the height of the tower?
- If tan A = 5/12, find the value of sin A + cos A.
- The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude is 30ยฐ than when it was 60ยฐ. Find the height of the tower.
๐ Conclusion
Trigonometry is a foundational topic in mathematics with numerous real-world applications. Understanding the basic trigonometric ratios and their relationships will provide a solid base for more advanced mathematical concepts.
