Trigonometric functions of an acute angle in a right triangle

In this publication, we will consider the ratios of the legs and hypotenuse in a right triangle, expressed as trigonometric functions of an acute angle, and also learn how to apply the knowledge gained in practice to solve problems.

Side ratios in a right triangle

Let’s say we have a triangle (rectangular) with sides a, b, c and an acute angle α.

The following is true for him:

1. Sine of the angle α equal to the ratio of the opposite leg to the hypotenuse:

   without α = b/c

2. Cosine of an angle α is equal to the ratio of the adjacent leg to the hypotenuse:

   cos α = a/c

3. Take a tangent α equals the ratio of the opposite leg to the adjacent:

   tg α = b/a

4. Cotangent of an angle α equals the ratio of the adjacent leg to the opposite:

   ctg α = a/b

5. Secant of an angle α defined as the ratio of the hypotenuse to the adjacent leg:

   dry α = c/a

6. Cosecant of an angle α defined as the ratio of the hypotenuse to the opposite leg:

   cosec α = c/b

Examples of tasks

Task 1

In a right triangle, one of the legs is 3 cm, and the hypotenuse is 5 cm. Find the angle that is opposite the given leg.

Decision:

Let the unknown angle be α. We apply the sine formula to find it:

without α = 3 cm / 5 cm = 0,6. Therefore, the angle α = arcsin (0,6) ≈ 36,87°.

Task 2

In a rectangular one of the acute angles is 45 °, and the leg adjacent to it is 3 cm. Find the hypotenuse.

Decision:

Since we know the angle (α) and the adjacent leg (a), we derive the length of the hypotenuse from the cosine formula (c): c = a / cos α = 3 cm / cos 45° ≈ 4,24 cm.