TutorChase logo
CIE A-Level Maths Study Notes

1.5.2 Exact Values of Trigonometric Ratios

Trigonometry, an integral part of mathematics, delves into the relationships between angles and sides of triangles. This section elaborates on the exact values of sine, cosine, and tangent for angles of 30°, 45°, and 60°. These angles are crucial due to their regular appearance in mathematical problems and real-world applications.

Understanding Trigonometric Functions

Trigonometric functions are fundamental in understanding the dynamics of angles and lengths in a circle, especially the unit circle.

Sine Function (sin)

  • Definition: In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse.
  • Unit Circle Representation: The y-coordinate of a point on the unit circle.
  • Significance: The sine function is pivotal in calculating heights and distances in various fields, including engineering and physics.

Cosine Function (cos)

  • Definition: In a right-angled triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
  • Unit Circle Representation: The x-coordinate of a point on the unit circle.
  • Significance: Cosine is used extensively in waveform analysis and in determining the phase and amplitude of oscillations.

Tangent Function (tan)

  • Definition: In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the adjacent side.
  • Unit Circle Representation: The ratio of the y-coordinate to the x-coordinate of a point on the unit circle.
  • Significance: Tangent is crucial in surveying, navigation, and in the study of periodic functions.

Special Angles and Their Trigonometric Values

Understanding special triangles, such as the 30-60-90 and 45-45-90 triangles, is crucial for comprehending trigonometric ratios.

Trigonometric ratios

Image courtesy of OnlineMath4All

unit circle

Image courtesy of Clip Art ETC


Example 1: Using Sine Function

Calculate the height of a building given an angle of elevation of 30° from a point 100 meters away.

Solution:

  1. Use the sine function: sin(30°)=Height of the building100 m\sin(30°) = \frac{\text{Height of the building}}{100 \text{ m}}.
  2. Knowing sin(30°)=12\sin(30°) = \frac{1}{2}, solve for the height: Height of the building=12×100 m\text{Height of the building} = \frac{1}{2} \times 100 \text{ m}.
  3. The height of the building is 50 m.

Example 2: Using Cosine Function

Find the length of the base of a right triangle with a hypotenuse of 10 cm and an angle of 60°.

Solution:

  1. Use the cosine function: cos(60°)=Base length10 cm.\cos(60°) = \frac{\text{Base length}}{10 \text{ cm}}.
  2. Knowing cos(60°)=12\cos(60°) = \frac{1}{2}, solve for the base length: Base length=12×10 cm\text{Base length} = \frac{1}{2} \times 10 \text{ cm}.
  3. The base length is 5 cm.

Example 3: Using Tangent Function

Determine the angle θ made by a 10-meter ladder with the ground, where the foot of the ladder is 6 meters away from the wall.

Solution:

  1. Use the tangent function: tan(θ)=10 m6 m\tan(\theta) = \frac{10 \text{ m}}{6 \text{ m}}.
  2. Find the angle using the inverse tangent function: θ=tan1(10 m6 m)\theta = \tan^{-1}\left(\frac{10 \text{ m}}{6 \text{ m}}\right).
  3. The angle θ is approximately 59.04°.
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

Hire a tutor

Please fill out the form and we'll find a tutor for you.

1/2 About yourself
Still have questions?
Let's get in touch.