Sample · 01-Trigonometry
Chapter 1 Overview
Math 10 · CHAPTER 1: Trigonometry
Trigonometry means “triangle measurement”. In a right triangle, the ratios of the sides depend only on the angles, so one angle and one side are enough to find everything else. Topic 1.1 defines sine, cosine, tangent and cotangent in a right triangle. Topic 1.2 proves the identities \( \sin^{2}\alpha + \cos^{2}\alpha = 1 \), \( \tan\alpha \cdot \cot\alpha = 1 \) and the complementary-angle rules. Topic 1.3 finds the exact ratios of \( 30^{\circ} \), \( 45^{\circ} \) and \( 60^{\circ} \). Topic 1.4 writes the ratios in terms of each other. Topic 1.5 moves the angle onto the unit circle, where cosine and sine become coordinates and angles up to \( 180^{\circ} \) make sense. Topic 1.6 gives angles a direction, and Topic 1.7 introduces the auxiliary elements of a triangle: angle bisectors, perpendicular bisectors, altitudes and medians. The chapter ends with SAT - ACT questions, a challenge zone of competition and olympiad problems, and four chapter exams.
🎓 Where This Chapter Can Take You
Deniz studied Geomatics Engineering at Istanbul Technical University and then Geomatics at ETH Zurich. Her team had to find the height of an old clock tower without climbing it. She stood \( 20 \) m from the base of the tower and aimed her theodolite at the top: the angle of elevation was \( 60^{\circ} \).
The tower, the ground and her line of sight form a right triangle. The side opposite the \( 60^{\circ} \) angle is the height \( h \), and the side next to it is \( 20 \) m, so \( \tan 60^{\circ} = \tfrac{h}{20} \). Since \( \tan 60^{\circ} = \sqrt{3} \), the height is \( h = 20\sqrt{3} \approx 34.6 \) m. No ladder, no tape measure up the wall — only one angle and one distance. The table shows where the ideas of this chapter are used.
| Career | What they do | Topics used | Example university programs |
|---|---|---|---|
| Geomatics engineer | Measures heights and distances that cannot be reached | 1.1 | Geomatics Engineering, Istanbul Technical University; Geomatics, ETH Zurich |
| Electronics engineer | Checks tilt sensors with \( \sin^{2} + \cos^{2} = 1 \) | 1.2 | Electrical and Electronics Engineering, METU; Electrical Engineering, TU Munich |
| Air traffic controller | Finds the altitude of an aircraft from an angle | 1.3 | Air Traffic Control, Anadolu University; Air Transport Management, Cranfield University |
| Civil engineer | Designs roads and ramps with a given slope percentage | 1.4 | Civil Engineering, Bogazici University; Civil Engineering, Imperial College London |
| Game developer | Rotates characters and cameras around a point | 1.5 | Computer Engineering, Bilkent University; Computer Games Technology, Abertay University |
| Robotics engineer | Programs clockwise and counterclockwise turns | 1.6 | Mechatronics Engineering, Sabanci University; Robotics Engineering, Worcester Polytechnic Institute |
| Architect | Places a point at equal distances from three buildings | 1.7 | Architecture, METU; Architecture, TU Delft |
1.1 Trigonometric Ratios in a Right Triangle - Part 1
1.1 Trigonometric Ratios in a Right Triangle - Part 1
💼 Math at Work — Geomatics Engineer
Continue reading — 2 chapters, worked examples and practice.
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