Using trigonometric relationships

At trigonometric relations are formulas that relate the angles and sides of a right triangle. These formulas involve the functions sine, cosine and tangentand have many applications in geometric problems involving this type of triangle.

Trigonometric relations in the right triangle

O right triangle it is the triangle that has a right angle (90°) and two acute angles (less than 90°). The sides of the right triangle are called the hypotenuse and sides, and the sides can be opposite or adjacent, depending on the reference angle.

rectangle triangle

Elements of the right triangle:

Hypotenuse: side opposite right angle;
Opposite side: side opposite the considered acute angle;
Adjacent side: side consecutive to the considered acute angle.

Formulas:

considering the angle $\dpi{120} \alpha$ of the right triangle, we have to:

$\dpi{120} \mathbf{sen\, \boldsymbol{\alpha} = \frac{catheto\, opposite}{hypotenuse}}$

$\dpi{120} \mathbf{cos\, \boldsymbol{\alpha} = \frac{catheto\, adjacent}{hypotenuse}}$

$\dpi{120} \mathbf{tan\, \boldsymbol{\alpha} = \frac{side\, opposite}{side \, adjacent}}$

Note: The hypotenuse of the right triangle is always the same, the opposite and adjacent sides vary in relation to the acute angle under consideration.

Examples - Using Trigonometric Relationships

Below are examples of how to use trigonometric relationships.

instagram story viewer

Example 1: Calculate the value of x and y in the triangle below:

triangle

From the sine of the 30° angle, we can determine the value of x, which is the hypotenuse of the triangle.

$\dpi{120} \mathrm{sen\, 30^{\circ} =\frac{5}{x}}$

$\dpi{120} \Rightarrow \mathrm{ x=\frac{5}{sen\, 30^{\circ}}}$

Check out some free courses

Free Online Inclusive Education Course
Free Online Toy Library and Learning Course
Free Online Math Games Course in Early Childhood Education
Free Online Pedagogical Cultural Workshops Course

$\dpi{120} \mathrm{\Rightarrow x = 10}$

Now, one of the ways to find the value of y is from the cosine of the 30° angle. In this case, y is the leg adjacent to the 30° angle.

$\dpi{120} \mathrm{cos\, 30^{\circ} =\frac{y}{10}}$

$\dpi{120} \Rightarrow \mathrm{ y = 10\cdot cos\, 30^{\circ}}$

$\dpi{120} \Rightarrow \mathrm{ y \approx 9}$

Example 2: Determine the measure of the angles $\dpi{120} \alpha$ and $\dpi{120} \beta$ from the triangle below:

triangle

First, let's determine the angle $\dpi{120} \alpha$ :

$\dpi{120} \mathrm{sen\, \alpha = \frac{5}{6,4}}$

$\dpi{120} \mathrm{\Rightarrow \alpha = sen^{-1} \left ( \frac{5}{6,4}\right )}$

$\dpi{120} \mathrm{\Rightarrow \alpha \approx 51.37^{\circ}}$

Now let's determine the angle $\dpi{120} \beta$ :

$\dpi{120} \mathrm{sen\, \beta = \frac{4}{6,4}}$

$\dpi{120} \mathrm{\Rightarrow \beta = sen^{-1} \left ( \frac{4}{6,4}\right )}$

$\dpi{120} \Rightarrow \beta \approx 38.68$

Note that we used sine in both cases, but we could also use cosine and arrive at these same results.

You may also be interested:

trigonometric table
trigonometric circle
Derived relationships
List of Trigonometry Exercises
Sine and Cosine of Obtuse Angles

The password has been sent to your email.

Female hormones and the menstrual cycle

Female hormones and the menstrual cycle

Hormones can be produced by glands in the endocrine system or by specialized neurons. These chemi...

3 Tips to Help Insecure Children

Every father and mother want their child to be happy and successful in life, however, these expec...

The Brazilian People and the Proclamation of the Republic of Brazil

THE Proclamation of the Republic of Brazilit is one of the most important phases in our recent hi...