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

The basic trigonometric ratios, sine, cosine and tangent, compare the sides of a right-angled triangle. Their reciprocals are cosecant, secant and cotangent. For angles beyond the first quadrant, each ratio equals plus or minus the ratio of an acute angle, and the sign depends on the quadrant.

Behavioural objectives

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

• define sinӨ, cosӨ and tanӨ in terms of the sides of a right-angled triangle;
• define secӨ, cosecӨ and cotӨ as the reciprocals of the basic ratios;
• use Pythagoras’ theorem to find all six ratios when one ratio is given;
• express the ratios of angles in the second, third and fourth quadrants in terms of acute angles;
• state which ratios are positive in each quadrant.

What are the basic trigonometric ratios?

The basic trigonometric ratios can be defined in terms of the sides of a right-angled triangle.

For example, in triangle PQR, angle QPR = Ө and angle PRQ = 90°. Let p = QR, q = PR and r = PQ. We define the three basic ratios as follows:

• Cosine of angle Ө = PR / PQ = q / r
• Sine of angle Ө = QR / PQ = p / r
• Tangent of angle Ө = QR / PR = p / q

The cosine of angle Ө, sine of angle Ө and tangent of angle Ө are abbreviated as cosӨ, sinӨ and tanӨ respectively.

Thus: cosӨ = q / r, sinӨ = p / r, tanӨ = p / q.

Also, tanӨ = p / q = (p / r) ÷ (q / r) = sinӨ / cosӨ.

What are the reciprocals of the basic ratios?

We define the reciprocals of the three basic ratios as follows:

• Secant of angle Ө = PQ / PR = r / q = 1 / cosine of angle Ө
• Cosecant of angle Ө = PQ / QR = r / p = 1 / sine of angle Ө
• Cotangent of angle Ө = PR / QR = q / p = 1 / tangent of angle Ө

The secant, cosecant and cotangent of angle Ө are abbreviated secӨ, cosecӨ and cotӨ respectively.

• secӨ = r / q = 1 / cosӨ
• cosecӨ = r / p = 1 / sinӨ
• cotӨ = 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

Use Pythagoras’ theorem to find PR.

PQ² = PR² + QR²
13² = PR² + 5²
PR² = 13² – 5²
= 169 – 25
= 144
PR = 12

Thus, q = 12, r = 13, p = 5.

• 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

How do we find the ratios of a general angle?

First quadrant

sinӨ = y
cosӨ = x
tanӨ = y / x

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

Solution:

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

Second quadrant

sin (180° – Ө) = sinӨ
cos (180° – Ө) = -cosӨ
tan (180° – Ө) = -tanӨ

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

Solution:

• 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

sin (180° + Ө) = -sinӨ
cos (180° + Ө) = -cosӨ
tan (180° + Ө) = tanӨ

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

Solution:

• 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

sin (360° – Ө) = -sinӨ
cos (360° – Ө) = cosӨ
tan (360° – Ө) = -tanӨ

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

Solution:

• 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?

1. In the first quadrant, all the ratios are positive.
2. In the second quadrant, only the sine ratio is positive, while the rest are negative.
3. In the third quadrant, only the tangent ratio is positive, while the rest are negative.
4. In the fourth quadrant, only the cosine ratio is positive, while the rest are negative.

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°.