11.3 Introduction to Trigonometry

Introduction to Trigonometry

The NTS Geometry block opens with "Introduction to Trigonometry" — and because calculators are barred from the hall, every trigonometry item on the GAT General is built from standard angles and identities you can evaluate by hand. That constraint is good news: the topic is finite and highly learnable.

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The Three Primary Ratios

In a right triangle, relative to an acute angle $\theta$:

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$

The reciprocals are cosecant $= 1/\sin$, secant $= 1/\cos$, and cotangent $= 1/\tan$. Note the mismatch that costs marks: secant pairs with cosine, and cosecant pairs with sine — not the other way round.

The hypotenuse is always the side opposite the right angle. Opposite and adjacent are assigned relative to the angle you are working with, so they swap when you switch angles. In a 3-4-5 triangle with the angle facing the side of length 3: $\sin = 3/5$, $\cos = 4/5$, $\tan = 3/4$. Facing the side of length 4, those become $4/5$, $3/5$ and $4/3$.

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The Standard Angle Table

Memorise this grid. It answers the majority of GAT trigonometry items outright.

How to reconstruct it if you blank. Write $0, 1, 2, 3, 4$ across the sine row, divide each by 4, and take the square root: $0, \tfrac12, \tfrac{1}{\sqrt2}, \tfrac{\sqrt3}{2}, 1$. The cosine row is the same list reversed. The tangent row is sine divided by cosine.

Complementary Angles

$\sin\theta = \cos(90° - \theta) \qquad \tan\theta = \cot(90° - \theta) \qquad \sec\theta = \csc(90° - \theta)$

So $\sin 70° = \cos 20°$, and $\sin^2 40° + \sin^2 50° = \sin^2 40° + \cos^2 40° = 1$ — a favourite one-line item.

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The Pythagorean Identities

$\sin^{2}\theta + \cos^{2}\theta = 1 \qquad 1 + \tan^{2}\theta = \sec^{2}\theta \qquad 1 + \cot^{2}\theta = \csc^{2}\theta$

All three come from Pythagoras applied to a right triangle of hypotenuse 1; the second and third follow by dividing the first through by $\cos^{2}\theta$ and $\sin^{2}\theta$.

Worked example. If $\sin\theta = \frac{5}{13}$ and $\theta$ is acute, find $\tan\theta$.

$\cos^{2}\theta = 1 - \frac{25}{169} = \frac{144}{169} \;\Rightarrow\; \cos\theta = \frac{12}{13} \;\Rightarrow\; \tan\theta = \frac{5/13}{12/13} = \frac{5}{12}$

Recognising the 5-12-13 triple would have given this instantly. The triples worth knowing are 3-4-5, 5-12-13, 8-15-17 and 7-24-25.

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Radian Measure

$\pi \text{ radians} = 180° \qquad\Rightarrow\qquad 1 \text{ radian} = \frac{180°}{\pi} \approx 57.3°$

In radians the arc length and sector area formulas simplify to $s = r\theta$ and $A = \tfrac{1}{2}r^{2}\theta$.

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Angles of Elevation and Depression

The angle of elevation is measured upward from the horizontal to an object above; the angle of depression is measured downward from the horizontal to an object below. Because the two horizontals are parallel, the angle of elevation from the ground to a tower top equals the angle of depression from the tower top to that ground point — alternate angles.

Worked example 1. The angle of elevation of the top of a minaret from a point 60 m away on level ground is 30°. Find its height.

$\tan 30° = \frac{h}{60} \;\Rightarrow\; h = 60 \times \frac{1}{\sqrt{3}} = \frac{60}{\sqrt 3} = 20\sqrt{3} \approx 34.6\text{ m}$

Worked example 2. A 15 m ladder leans against a wall at 60° to the ground. How far up the wall does it reach, and how far is its foot from the wall?

$\text{height} = 15\sin 60° = 15 \times \frac{\sqrt 3}{2} \approx 12.99\text{ m} \qquad \text{base} = 15\cos 60° = 7.5\text{ m}$

Worked example 3. From the top of a 45 m tower, the angle of depression of a car is 45°. How far is the car from the base?

Since $\tan 45° = 1$, the horizontal distance equals the height: 45 m. Whenever 45° appears, the two legs are equal and no arithmetic is needed.

Setup discipline: sketch the triangle and label which side you know and which you want before choosing a ratio. Known adjacent and wanted opposite means tangent; known hypotenuse and wanted opposite means sine. Choosing the ratio after labelling, rather than before, removes almost every error on this topic.