In this Mathematics tutorials, we will study What is trigonometry? What is trigonometry used for? and few solved trigonometry problems. Trigonometry is a study of the relation between the sides of the angles of a triangle. Hence, trigonometry is also useful for general triangles, not just the right-angled triangle.
To understand the deep roots of trigonometry we need to understand from it was derived. This word Trigonometry came into existence in the 16th century from the Greek words ‘trigono'(which means triangle) and ‘metron’ (is to measure). In simple terms, if we know the two sides of a right-angle triangle then thirds sided can be calculated using the Pythagoras theorem.
Let’s become the expert triangle expert by solving few problem questions of trigonometry.
What is trigonometry used for?
We have understood what is trigonometry. Now, we will try to focus on What is trigonometry used for? Its application in real life.
Real-life applications of trigonometry
-To measure the height of the mountain and building heights using trigonometry
-Videos games use trigonometry
– Construction of building, bridges use trigonometry
– Physics use trigonometry
-Criminology and archaeologists used trigonometry
– Marine bio and marine engineering uses trigonometry
-Navigators use trigonometry
– To measure the heights of the tides the oceanographer uses trigonometry
-The sound and light waves uses the fundamentals of trigonometry like sine and cosine to the theory of periodic functions.
-In fact, calculus uses trigonometry and algebra
-To make roof of a house inclined the engineers use trigonometry
-Aviation and navel industry also use trigonometry
-Most importantly, Cartography use trigonometry (to create maps)
Trigonometry Problems
To understand the below question you must know the Trigonometry formulas. For that, please check our blog on the trigonometry table.
Problem 1: Prove \sin^2 \frac{\pi}{6} +\cos^2\frac{\pi}{3}-\tan^2\frac{\pi}{4}=-\frac{1}{2}
Solution: We have,
LHS=\sin^2 \frac{\pi}{6} +\cos^2\frac{\pi}{3}-\tan^2\frac{\pi}{4}=(\frac{1}{2})^2+(\frac{1}{2})^2-(1)^2\\ =\frac{1}{4}+\frac{1}{4}-1\\ =\frac{1+1-4}{4}\\ =-\frac{1}{2}=RHS
Problem 2: Evaluate \sin 15°
Solution: We have, \sin 15°=\sin (45°-30°) Such that, \sin (45°-30°)= \sin 45° \cos 30°- \cos 45° \sin 30° (Using trigonometry formula i.e \sin(a-b) = \sin(a)\cos(b)–\cos(a)\sin(b)) \\ =(\frac{1}{\surd 2})(\frac{\surd 3}{2})-(\frac{1}{\surd 2})(\frac{1}{2})\\ =\frac{\surd 3 -1}{2\surd 2}
Problem 3: Prove that \tan 15 ° +cot 15° =4
Solution We have,
\tan 15° = \frac{\tan 45° - \tan 30°}{1+\tan 45° \tan 30° }=\frac{\surd 3-1}{\surd 3+1}\\ \cot 15°=\frac{1}{\tan 15°}=\frac{\surd 3+1}{\surd 3-1}\\ \therefore \tan 15 ° +cot 15° =\frac{\surd 3-1}{\surd 3+1}+\frac{\surd 3+1}{\surd 3-1}\\ =\frac{(\surd 3-1)^2+(\surd 3+1)^2}{(\surd 3+1)(\surd 3-1)}\\ =\frac{3+1-2\surd 3+3+1+2\surd 3}{3-1}\\ =\frac{8}{2}\\ =4
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