Trigonometric Ratio of Special Angles and Logical Reasoning – SS1 Further Mathematics Lesson Note (Second Term)

Trigonometric ratios compare the sides of a right-angled triangle, and the ratios of special angles (30°, 45° and 60°) have exact values that can be worked out without a table.

Behavioural objectives

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

1. define sinӨ, cosӨ and tanӨ in terms of the sides of a right-angled triangle;
2. define secӨ, cosecӨ and cotӨ as the reciprocals of the basic ratios;
3. use Pythagoras’ theorem to find all six ratios when one ratio is given;
4. express the ratios of angles in the second, third and fourth quadrants in terms of acute angles;
5. state which ratios are positive in each quadrant;
6. state the exact values of the sine, cosine and tangent of 30°, 45° and 60° and use them to solve problems.

What are the basic trigonometric ratios?

The basic trigonometric ratios are sine, cosine and tangent, which compare two sides of a right-angled triangle.

Labelling the triangle

• Triangle PQR: angle QPR = Ө and angle PRQ = 90°.
• p: the side QR, opposite angle Ө.
• q: the side PR, next to angle Ө (adjacent).
• r: the side PQ, opposite the right angle (hypotenuse).

Definitions

• Sine: sinӨ = QR / PQ = p / r (opposite ÷ hypotenuse).
• Cosine: cosӨ = PR / PQ = q / r (adjacent ÷ hypotenuse).
• Tangent: tanӨ = QR / PR = p / q (opposite ÷ adjacent).

• The words sine, cosine and tangent are abbreviated sinӨ, cosӨ and tanӨ.
• Also, tanӨ = p / q = (p / r) ÷ (q / r) = sinӨ / cosӨ.

What are the reciprocals of the basic ratios?

The reciprocals of the basic ratios are secant, cosecant and cotangent, each found by turning a basic ratio upside down.

• Secant: secӨ = PQ / PR = r / q = 1 / cosӨ.
• Cosecant: cosecӨ = PQ / QR = r / p = 1 / sinӨ.
• Cotangent: cotӨ = PR / QR = q / p = 1 / tanӨ = cosӨ / sinӨ.

Example 1

Given that sinӨ = 5 / 13 and Ө is acute, find: (1) cosӨ, (2) tanӨ, (3) secӨ, (4) cosecӨ, (5) cotӨ.

Solution

1. Write sinӨ = p / r = 5 / 13, so p = 5 and r = 13.
2. Use Pythagoras’ theorem to find PR: PQ² = PR² + QR².
3. 13² = PR² + 5², so PR² = 169 – 25 = 144.
4. PR = 12, so q = 12.

• cosӨ = q / r = 12 / 13
• tanӨ = p / q = 5 / 12
• secӨ = r / q = 13 / 12
• cosecӨ = r / p = 13 / 5
• cotӨ = q / p = 12 / 5

What are the trigonometric ratios of special angles?

The special angles are 30°, 45° and 60°, whose ratios can be written exactly using square roots.

How do we get the ratios of 45°?

1. Draw a right-angled isosceles triangle with two equal sides of 1 unit.
2. Its two acute angles are both 45°.
3. The hypotenuse is √(1² + 1²) = √2.
4. Read the ratios: sin 45° = 1 / √2, cos 45° = 1 / √2, tan 45° = 1.

How do we get the ratios of 30° and 60°?

1. Draw an equilateral triangle of side 2 units. Each angle is 60°.
2. Draw a line from one vertex to the middle of the opposite side.
3. This splits the triangle into two right-angled triangles with angles 30°, 60° and 90°.
4. The short side is 1, the hypotenuse is 2, and the height is √(2² – 1²) = √3.
5. Read the ratios from one half-triangle, as in the table below.

Table of special angles

• Note: 1 / √2 can also be written as √2 / 2, and 1 / √3 as √3 / 3.
• Complementary angles: sin 30° = cos 60° and sin 60° = cos 30°.

Example 2

Evaluate sin 60° × cos 30° + tan 45°, without a table.

1. sin 60° = √3 / 2 and cos 30° = √3 / 2.
2. Their product = (√3 / 2) × (√3 / 2) = 3 / 4.
3. tan 45° = 1.
4. Sum = 3 / 4 + 1 = 7 / 4.

How do we find the ratios of a general angle?

For an angle beyond 90°, the ratio equals plus or minus the ratio of an acute angle, and the sign depends on the quadrant.

• In the first quadrant, a point on the unit circle has coordinates (x, y).
• Then sinӨ = y, cosӨ = x and tanӨ = y / x.

What are the rules for each quadrant?

First quadrant

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

• sin 37° = 0.6018
• cos 75° = 0.2588
• tan 62° = 1.881

Second quadrant

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

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

Third quadrant

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

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

Fourth quadrant

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

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

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.

How do we reason out the ratio of any angle?

To find the ratio of any angle, find its quadrant, then its acute angle, then the sign.

1. Find the quadrant in which the angle lies.
2. Find the acute angle with the x-axis: 180° – Ө, Ө – 180° or 360° – Ө.
3. Write the ratio of that acute angle, using a table or the special-angle values.
4. Choose the sign from the quadrant: All, Sine, Tangent, Cosine.

Example 3

Find the exact values of (a) sin 150° (b) cos 225° (c) tan 300°.

• sin 150° is in the second quadrant, so sin 150° = sin 30° = 1 / 2.
• cos 225° is in the third quadrant, so cos 225° = -cos 45° = -1 / √2.
• tan 300° is in the fourth quadrant, so tan 300° = -tan 60° = -√3.

Evaluation

1. Write down the definitions of sinӨ, cosӨ and tanӨ in terms of p, q and r.
2. Write secӨ, cosecӨ and cotӨ in terms of cosӨ, sinӨ and tanӨ.
3. Given that sinӨ = 5 / 13 and Ө is acute, state the value of cotӨ.
4. Express sin 143° and cos 236° in terms of acute angles.
5. In which quadrants is the tangent ratio positive?

Assignment

1. Given that cosӨ = 4 / 5 and Ө is acute, use Pythagoras’ theorem to find sinӨ, tanӨ, secӨ, cosecӨ and cotӨ.
2. Express each of the following in terms of an acute angle: (a) sin 125° (b) cos 200° (c) tan 300°.
3. Without a table, evaluate sin 30° × cos 60° + tan 60° × tan 30°.