Trigonometric Functions – SS2 Further Mathematics Lesson Note (First Term)

Trigonometric ratios are ratios of the sides of a right-angled triangle, used to describe an angle θ.

Behavioural objectives

By the end of the lesson, students should be able to:

1. define sine, cosine and tangent of an angle using the sides of a right-angled triangle;
2. define secant, cosecant and cotangent as reciprocals of the basic ratios;
3. find all six ratios of an acute angle when one ratio is given;
4. express the ratio of an angle in the second, third or fourth quadrant in terms of an acute angle;
5. state which ratios are positive in each quadrant.

What are the basic trigonometric ratios?

The three basic trigonometric ratios are sine, cosine and tangent, each defined by two sides of a right-angled triangle.

The triangle used for the definitions

• Triangle PQR: right-angled at R, so angle PRQ = 90°.
• Angle θ: the angle at P, so angle QPR = θ.
• p = QR: the side opposite θ.
• q = PR: the side adjacent to θ.
• r = PQ: the hypotenuse, the longest side, opposite the right angle.

Definitions of the basic ratios

• Sine of θ (sinθ): opposite over hypotenuse, so sinθ = QR/PQ = p/r.
• Cosine of θ (cosθ): adjacent over hypotenuse, so cosθ = PR/PQ = q/r.
• Tangent of θ (tanθ): opposite over adjacent, so tanθ = QR/PR = p/q.

How is tanθ related to sinθ and cosθ?

Tangent of θ equals sine of θ divided by cosine of θ.

1. Write sinθ/cosθ = (p/r) ÷ (q/r).
2. Multiply by the reciprocal: (p/r) × (r/q).
3. Cancel r to get p/q.
4. Since p/q = tanθ, tanθ = sinθ/cosθ.

What are the reciprocals of the basic ratios?

The reciprocal ratios are secant, cosecant and cotangent, found by turning each basic ratio upside down.

• Secant (secθ): hypotenuse over adjacent, so secθ = PQ/PR = r/q = 1/cosθ.
• Cosecant (cosecθ): hypotenuse over opposite, so cosecθ = PQ/QR = r/p = 1/sinθ.
• Cotangent (cotθ): adjacent over opposite, so cotθ = PR/QR = q/p = 1/tanθ = cosθ/sinθ.

Summary of the six ratios

How do we find all six ratios when one ratio is given?

Use the given ratio to label two sides of the triangle, find the third side by Pythagoras’ theorem, then write every ratio.

Steps

1. Draw triangle PQR and mark angle θ at P.
2. Label the two sides given by the ratio.
3. Find the third side using PQ² = PR² + QR².
4. Write the six ratios from the three sides.

Example 1

Given that sinθ = 5/13 and θ is acute, find cosθ, tanθ, secθ, cosecθ and cotθ.

1. sinθ = p/r = 5/13, so p = 5 and r = 13.
2. PQ² = PR² + QR², so 13² = PR² + 5².
3. PR² = 169 – 25 = 144.
4. PR = 12, so q = 12.

• cosθ: q/r = 12/13.
• tanθ: p/q = 5/12.
• secθ: r/q = 13/12.
• cosecθ: r/p = 13/5.
• cotθ: q/p = 12/5.

What are the ratios of general angles?

General angles are angles beyond 90°, and their ratios are written in terms of an acute angle using the quadrant they lie in.

• First quadrant: angles from 0° to 90°.
• Second quadrant: angles from 90° to 180°.
• Third quadrant: angles from 180° to 270°.
• Fourth quadrant: angles from 270° to 360°.

First quadrant

In the first quadrant, all ratios are read directly from the tables.

• On a unit circle: sinθ = y, cosθ = x and tanθ = y/x for a point (x, y).

Example: Use the table to evaluate (a) sin 37° (b) cos 75° (c) tan 62°.

• (a) sin 37° = 0.6018
• (b) cos 75° = 0.2588
• (c) tan 62° = 1.881

Second quadrant

For an angle between 90° and 180°, use 180° – θ as the acute angle.

• Sine: sin (180° – θ) = sinθ.
• Cosine: cos (180° – θ) = -cosθ.
• Tangent: tan (180° – θ) = -tanθ.

Example: Use the table to evaluate (a) sin 143° (b) cos 115° (c) tan 125°.

• (a) sin 143° = sin (180° – 143°) = sin 37° = 0.6018
• (b) cos 115° = -cos (180° – 115°) = -cos 65° = -0.4226
• (c) tan 125° = -tan (180° – 125°) = -tan 55° = -1.428

Third quadrant

For an angle between 180° and 270°, subtract 180° to get the acute angle.

• Sine: sin (180° + θ) = -sinθ.
• Cosine: cos (180° + θ) = -cosθ.
• Tangent: tan (180° + θ) = tanθ.

Example: Use the table to evaluate (a) sin 220° (b) cos 236° (c) tan 242°.

• (a) sin 220° = sin (180° + 40°) = -sin 40° = -0.6428
• (b) cos 236° = cos (180° + 56°) = -cos 56° = -0.5592
• (c) tan 242° = tan (180° + 62°) = tan 62° = 1.881

Fourth quadrant

For an angle between 270° and 360°, use 360° – θ as the acute angle.

• Sine: sin (360° – θ) = -sinθ.
• Cosine: cos (360° – θ) = cosθ.
• Tangent: tan (360° – θ) = -tanθ.

Example: Use the table to evaluate (a) sin 310° (b) cos 285° (c) tan 334°.

• (a) sin 310° = -sin (360° – 310°) = -sin 50° = -0.7660
• (b) cos 285° = cos (360° – 285°) = cos 75° = 0.2588
• (c) tan 334° = -tan (360° – 334°) = -tan 26° = -0.4877

Summary of the quadrant formulas

Which ratios are positive in each quadrant?

All ratios are positive in the first quadrant, and in the other quadrants only one of sine, cosine and tangent is positive.

1. First quadrant: all the ratios are positive.
2. Second quadrant: only sine is positive, and the rest are negative.
3. Third quadrant: only tangent is positive, and the rest are negative.
4. Fourth quadrant: only cosine is positive, and the rest are negative.

Sign table

• Reciprocal rule: a ratio and its reciprocal always have the same sign.
• Memory aid: “All Students Take Cocoa” gives All, Sin, Tan, Cos from the first to the fourth quadrant.

Evaluation

1. Write secθ, cosecθ and cotθ in terms of p, q and r for triangle PQR.
2. Given that sinθ = 3/5 and θ is acute, find cosθ and tanθ.
3. Write sin 143° and cos 115° in terms of an acute angle.
4. State the quadrants in which the tangent ratio is positive.

Assignment

1. Given that cosθ = 4/5 and θ is acute, find sinθ, tanθ, secθ, cosecθ and cotθ. Then express (a) sin 150° (b) cos 200° (c) tan 300° in terms of an acute angle.