≡Mathematics EN 9th grade: 1. What is Trigonometry
1. What is Trigonometry
Trigonometry studies the relationships between angles and sides of right triangles.
✔ It applies to triangles with one angle of \( 90^\circ \)
2. Trigonometric ratios
In a right triangle:
Hypotenuse → longest side
Opposite side → side opposite the angle
Adjacent side → side next to the angle
✔ Sine
\( \sin(\theta) = \frac{\text{cateto oposto}}{\text{hipotenusa}} \)
✔ Cosine
\( \cos(\theta) = \frac{\text{cateto adjacente}}{\text{hipotenusa}} \)
✔ Tangent
\( \tan(\theta) = \frac{\text{cateto oposto}}{\text{cateto adjacente}} \)
3. Practical example
In a triangle:
opposite side = \( 3 \)
adjacent side = \( 4 \)
hypotenuse = \( 5 \)
👉 \( \sin(\theta) = \frac{3}{5} \)
👉 \( \cos(\theta) = \frac{4}{5} \)
👉 \( \tan(\theta) = \frac{3}{4} \)
4. Using the calculator
✔ Use degree mode (DEG)
📌 Example:
\( \sin(30^\circ) = 0.5 \)
\( \cos(60^\circ) = 0.5 \)
\( \tan(45^\circ) = 1 \)
✔ To find angles:
\( \sin^{-1}, \cos^{-1}, \tan^{-1} \)
5. Trigonometric table (important values)
\( \sin(30^\circ) = \frac{1}{2} \)
\( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
\( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)
\( \sin(45^\circ) = \frac{\sqrt{2}}{2} \)
\( \cos(45^\circ) = \frac{\sqrt{2}}{2} \)
\( \tan(45^\circ) = 1 \)
\( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
\( \cos(60^\circ) = \frac{1}{2} \)
6. Application: calculating distances
Trigonometry allows calculating distances without direct measurement.
📌 Example 1
A triangle with:
angle \( 30^\circ \)
hypotenuse = \( 10 \)
👉 \( \sin(30^\circ) = \frac{x}{10} \)
👉 \( 0.5 = \frac{x}{10} \)
👉 \( x = 5 \)
📌 Example 2
Height of a building:
angle = \( 45^\circ \)
distance to the building = \( 8 \)
👉 \( \tan(45^\circ) = \frac{h}{8} \)
👉 \( 1 = \frac{h}{8} \)
👉 \( h = 8 \)
\( \tan(60^\circ) = \sqrt{3} \)
Did you know?
