Trigonometric Functions of Acute Angles: Definitions, Properties, and Applications — 6.1

Trigonometric Functions of Acute Angles

Definition of Acute Angles and Right Triangles

An acute angle is an angle between 0° and 90°. In a right triangle, one angle is exactly 90°, and the other two are acute. Trigonometric functions relate the angles of a right triangle to the ratios of its sides.

• Opposite side: The side opposite the angle in question.

• Adjacent side: The side next to the angle in question (but not the hypotenuse).

• Hypotenuse: The longest side, opposite the right angle.

Trigonometric Ratios

The six basic trigonometric functions for an acute angle θ in a right triangle are defined as follows:

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Cosecant: $\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}}$

• Secant: $\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}}$

• Cotangent: $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$

Properties of Similar Right Triangles

All right triangles with the same acute angle are similar, so the trigonometric ratios depend only on the angle, not the size of the triangle.

• Triangles with the same acute angle have proportional sides.

• Trigonometric functions are constant for a given angle.

Reciprocal Functions

Each trigonometric function has a reciprocal:

• $\csc \theta = \frac{1}{\sin \theta}$

• $\sec \theta = \frac{1}{\cos \theta}$

• $\cot \theta = \frac{1}{\tan \theta}$

Trigonometric Values of Special Angles

Values for 0°, 30°, 45°, 60°, and 90° are commonly used and can be found using special right triangles (30-60-90 and 45-45-90 triangles).

Angle (°)	sin	cos	tan	csc	sec	cot
0	0	1	0	undefined	1	undefined
30	1/2	$\sqrt{3}/2$	$1/\sqrt{3}$	2	$2/\sqrt{3}$	$\sqrt{3}$
45	$1/\sqrt{2}$	$1/\sqrt{2}$	1	$\sqrt{2}$	$\sqrt{2}$	1
60	$\sqrt{3}/2$	1/2	$\sqrt{3}$	$2/\sqrt{3}$	2	$1/\sqrt{3}$
90	1	0	undefined	1	undefined	0

Function Values for Any Acute Angle

To find the trigonometric value for any acute angle, use a calculator or trigonometric tables. For example:

• $\sin 25^\circ \approx 0.4226$

• $\cos 25^\circ \approx 0.9063$

• $\tan 25^\circ \approx 0.4663$

Co-functions and Complements

Co-functions are pairs of trigonometric functions where the value of one at an angle equals the value of the other at the complement of that angle:

• $\sin(90^\circ - \theta) = \cos \theta$

• $\cos(90^\circ - \theta) = \sin \theta$

• $\tan(90^\circ - \theta) = \cot \theta$

• $\cot(90^\circ - \theta) = \tan \theta$

• $\sec(90^\circ - \theta) = \csc \theta$

• $\csc(90^\circ - \theta) = \sec \theta$

Applications of Trigonometric Functions

Trigonometric functions are used to solve problems involving right triangles, such as finding unknown side lengths or angles.

• Example: Given a right triangle with an angle of 35° and a hypotenuse of 10 units, find the length of the side opposite the angle:

  • $\sin 35^\circ = \frac{\text{opposite}}{10}$

  • Opposite $= 10 \times \sin 35^\circ \approx 5.74$ units

• Example: In a practical scenario, trigonometric functions can be used to determine the height of a building using the angle of elevation and the distance from the building.

Additional info:

• Trigonometric functions are foundational for further study in analytic geometry, calculus, and physics.

• Understanding the relationships between angles and side lengths is essential for solving real-world problems involving measurement and design.