Application of Trigonometry — Lesson
1) Hook — A Fun Real-Life Story
Imagine you are standing at the edge of the Ganges river in Varanasi, and you want to find the height of a tall temple tower across the river without crossing it. How can you do this using just a measuring tape and a protractor? This is where trigonometry comes to your rescue! By measuring the angle of elevation from your point to the top of the tower and the distance from you to the tower’s base, you can calculate the tower’s height accurately.
2) Core Concepts — Understanding Application of Trigonometry
Trigonometry helps us find unknown heights and distances in right-angled triangles by relating angles to side lengths.
Key Idea: In a right triangle, if you know one angle (other than 90°) and one side, you can find the other sides using trigonometric ratios:
| Trigonometric Ratio | Formula | Example |
|---|---|---|
| sin θ | Opposite side / Hypotenuse | sin 30° = 1/2 |
| cos θ | Adjacent side / Hypotenuse | cos 60° = 1/2 |
| tan θ | Opposite side / Adjacent side | tan 45° = 1 |
Example Problem: A flagpole stands vertically on the ground. From a point 20 m away from its base, the angle of elevation to the top of the pole is 30°. Find the height of the flagpole.
Solution:
- Let height of flagpole = h meters
- Distance from point to base = 20 m (adjacent side)
- Angle of elevation θ = 30°
Using tan θ = opposite / adjacent, we get:
tan 30° = h / 20
We know tan 30° = 1/√3, so
1/√3 = h / 20 ⇒ h = 20 / √3 = (20√3) / 3 ≈ 11.55 m
Therefore, the height of the flagpole is approximately 11.55 meters.
3) Key Formulas / Rules
Important Formulas to Remember:
- sin θ = Opposite / Hypotenuse
- cos θ = Adjacent / Hypotenuse
- tan θ = Opposite / Adjacent
- Height (h) = Distance × tan θ (when angle of elevation is given)
- Distance (d) = Height / tan θ (when height and angle of elevation are given)
4) Did You Know?
Trigonometry was first developed by Indian mathematician Aryabhata around 500 CE! He used sine tables to solve astronomical problems, laying the foundation for modern trigonometry.
5) Exam Tips — Avoid These Common Mistakes
- Always identify the right angle and the angle of elevation/depression carefully. Mislabeling leads to wrong side selection.
- Use the correct trigonometric ratio for the given sides. Remember: SOH-CAH-TOA helps recall sine, cosine, and tangent ratios.
- Keep your calculator in degree mode. Board exams usually provide angles in degrees.
- Write steps clearly. Partial marks are given for correct approach even if final answer is off.
- Practice problems involving angles of elevation and depression from NCERT and UP Board sample papers. These are frequently asked in exams.
Application of Trigonometry — Mcq
Application of Trigonometry — Mnemonic
Memorable Mnemonics for Application of Trigonometry (UP Board Class 10)
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Mnemonic 1: "Sine, Cosine, Tangent - Height & Distance Ka Funda!" 📏📐
Hindi rhyme to remember key concept:
“ऊँचाई नापनी हो, दूरी भी जाननी हो,
साइन, कोसाइन, टैन्जेंट से सब आसान हो!”Meaning: Use sine, cosine, tangent ratios to easily find heights and distances in problems.
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Mnemonic 2: "SOH-CAH-TOA, Height-Distance Ki Chhota Formula!" 🧮
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / AdjacentRemember: In height and distance problems, opposite side = height, adjacent side = distance from object.
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Mnemonic 3: "HDD - Height, Distance, Degree!" 🎯
Hindi phrase: “Height, Distance aur Degree ko yaad rakh, Trigonometry se sab solve kar!”
Tip: Always identify Height (H), Distance (D), and Angle of Elevation/Depression (Degree) first before applying formulas.
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