Introduction to Trigonometry · Class 10 Mathematics: 60 flashcards
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1. Which side is the hypotenuse?
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The side facing the right angle — it never changes with the angle chosen.
2. Which side is 'opposite'?
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The side facing the acute angle you are working with.
3. Define $\sin A$.
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$\frac{\text{opposite}}{\text{hypotenuse}}$
4. Define $\cos A$.
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$\frac{\text{adjacent}}{\text{hypotenuse}}$
5. Define $\tan A$.
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$\frac{\text{opposite}}{\text{adjacent}}$
6. What is $\operatorname{cosec} A$ the reciprocal of?
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$\sin A$.
7. What is $\sec A$ the reciprocal of?
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$\cos A$.
8. What is $\cot A$ the reciprocal of?
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$\tan A$.
9. Express $\tan A$ using sin and cos.
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$\tan A = \frac{\sin A}{\cos A}$
10. Why can't $\sin A$ exceed 1?
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The hypotenuse is the longest side, so the numerator cannot exceed the denominator.
11. What is the smallest possible value of $\sec A$?
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1 — it is the reciprocal of something at most 1.
12. Do $\tan A$ and $\cot A$ have an upper limit?
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No — the opposite side can be far longer than the adjacent one.
13. What does $\sin^2 A$ mean?
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$(\sin A)^2$ — the ratio squared.
14. What does $\sin^{-1} A$ mean?
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The inverse function: the angle whose sine is A.
15. Why do the ratios not depend on the triangle's size?
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Triangles with the same angles are similar, so their sides are in the same ratio.
16. Given $\sin A = \frac35$, find $\cos A$.
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The adjacent side is $\sqrt{25 - 9} = 4$, so $\cos A = \frac45$.
17. Given $\sin A = \frac35$, find $\cot A$.
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$\tan A = \frac34$, so $\cot A = \frac43$.
18. Given $\tan A = \frac12$, find the hypotenuse.
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$\sqrt{1 + 4} = \sqrt5$.
19. What are the three steps for finding all six ratios from one?
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Draw and label two sides; find the third by Pythagoras; read off the rest.
20. In a 3-4-5 triangle, what is $\operatorname{cosec} A$ if $\sin A = \frac35$?
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$\frac53$.
21. What is $\sin 30°$?
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$\frac12$
22. What is $\cos 30°$?
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$\frac{\sqrt3}{2}$
23. What is $\sin 45°$?
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$\frac{1}{\sqrt2}$
24. What is $\tan 45°$?
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1
25. What is $\sin 60°$?
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$\frac{\sqrt3}{2}$
26. What is $\cos 60°$?
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$\frac12$
27. What is $\tan 30°$?
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$\frac{1}{\sqrt3}$
28. What is $\tan 60°$?
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$\sqrt3$
29. What is $\sin 0°$ and $\cos 0°$?
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$\sin 0° = 0$ and $\cos 0° = 1$.
30. What is $\sin 90°$ and $\cos 90°$?
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$\sin 90° = 1$ and $\cos 90° = 0$.
31. What is $\tan 90°$?
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Undefined — the adjacent side is zero.
32. Which three standard values are undefined?
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$\tan 90°$, $\sec 90°$ and $\operatorname{cosec} 0°$ (and $\cot 0°$).
33. Where do the 45° values come from?
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A unit square cut along its diagonal: legs 1, 1 and hypotenuse $\sqrt2$.
34. Where do the 30° and 60° values come from?
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An equilateral triangle of side 2 cut in half, giving sides 1, $\sqrt3$, 2.
35. Give the memory trick for the sine row.
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$\sqrt{0/4}, \sqrt{1/4}, \sqrt{2/4}, \sqrt{3/4}, \sqrt{4/4}$.
36. How is the cosine row related to the sine row?
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It is the same list read backwards.
37. Does sin increase or decrease from 0° to 90°?
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It increases, from 0 to 1. Cosine decreases from 1 to 0.
38. Evaluate $\sin 30° \cos 60° + \cos 30° \sin 60°$.
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$\frac14 + \frac34 = 1$.
39. Is $\sin(A + B)$ equal to $\sin A + \sin B$?
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No. $\sin(30° + 60°) = 1$ but $\sin 30° + \sin 60° \approx 1.37$.
40. What is $\operatorname{cosec} 30°$?
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2 — the reciprocal of $\sin 30° = \frac12$.
41. State the first trigonometric identity.
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$\sin^2\theta + \cos^2\theta = 1$
42. State the second identity.
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$\sec^2\theta - \tan^2\theta = 1$
43. State the third identity.
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$\operatorname{cosec}^2\theta - \cot^2\theta = 1$
44. How is identity 1 proved?
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Take Pythagoras $p^2 + b^2 = h^2$ and divide every term by $h^2$.
45. How do you get identity 2 from identity 1?
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Divide every term by $\cos^2\theta$.
46. How do you get identity 3 from identity 1?
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Divide every term by $\sin^2\theta$.
47. What is the difference between an identity and an equation?
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An identity holds for every angle; an equation holds only for particular ones.
48. Write $\sin^2\theta$ in terms of $\cos\theta$.
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$\sin^2\theta = 1 - \cos^2\theta$
49. Write $\tan^2\theta$ in terms of $\sec\theta$.
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$\tan^2\theta = \sec^2\theta - 1$
50. Does $\sin^2\theta + \cos^2\theta = 1$ imply $\sin\theta + \cos\theta = 1$?
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No — at 45° the sum is $\sqrt2$.
51. What is the one rule for proving an identity?
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Work on ONE side only until it becomes the other.
52. What should you do when a proof stalls?
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Convert everything to sine and cosine.
53. Prove $\tan\theta + \cot\theta = \sec\theta \operatorname{cosec}\theta$.
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$\frac{\sin}{\cos} + \frac{\cos}{\sin} = \frac{\sin^2 + \cos^2}{\sin\cos} = \frac{1}{\sin\cos}$.
54. What is the conjugate of $\sec\theta - \tan\theta$?
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$\sec\theta + \tan\theta$.
55. Simplify $\frac{1}{\sec\theta - \tan\theta}$.
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$\sec\theta + \tan\theta$, since the denominator becomes $\sec^2 - \tan^2 = 1$.
56. What must you remember when using a conjugate?
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Multiply BOTH the numerator and the denominator.
57. Simplify $\sin^4\theta - \cos^4\theta$.
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$\sin^2\theta - \cos^2\theta$, since the other factor is 1.
58. For which angles does identity 2 fail?
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At 90°, where sec and tan are undefined.
59. For which angles does identity 3 fail?
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At 0°, where cosec and cot are undefined.
60. How should a proof answer end?
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With '= RHS, hence proved', having named each identity used.
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