Sine, cosine and tangent are elements that make up geometric knowledge. Knowing what it is for, how to use and calculate it is essential to obtain a comprehensive knowledge in trigonometry . The following text addresses this subject, I hope it will contribute to your learning.
Index
- What is sine, cosine and tangent?
- Sine, cosine and tangent formulas
- How to identify an opposite and an adjacent leg?
- Sine, cosine and tangent of notable angles
- Song remarkable angles
- Obtaining the values of the notable angles by means of mathematical calculations
- How to know when to use sine cosine and tangent
What is sine, cosine and tangent?
Sine, cosine and tangent are trigonometric ratios obtained through the relationships between the sides of a right triangle. Remember that this type of triangle has:
- An angle measuring 90 °.
- Two legs and a hypotenuse.
Looking at the figure, it is possible to notice that the hypotenuse is always opposite the 90 ° angle, and that the line segments that form the 90 ° angle are the sides.
Sine, cosine and tangent formulas
The general formulas of sine, cosine and tangent are described below:
- SINE
Description: Sine is the trigonometric ratio established in a right triangle between the opposite side and the hypotenuse.
- Cosine
Description: Cosine is the trigonometric ratio established in a right triangle between the adjacent leg and the hypotenuse.
- TANGENT
Description: Tangent is the trigonometric ratio established in a right triangle between the adjacent leg and the hypotenuse.
Obs . α can assume any value in degrees of the trigonometric cycle and can also assume values in π rad, that is, pi radians.
How to identify an opposite and an adjacent leg?
To answer this question, we must turn our eyes to the acute internal angles of the right triangle.
See that each angle was named with a Greek letter. The opposite and the adjacent leg for each angle are different line segments, but the hypotenuse will always be the same line segment.
For you to understand how to identify the opposite and the adjacent leg, observe the line segments that are used in each trigonometric ratio.
- Angle α
- Angle β
Sine, cosine and tangent of notable angles
The angles considered remarkable are: 30 °, 45 ° and 60 °. This is because these angles appear more frequently in trigonometric calculations.
Check in the table below the numerical values that these notable angles assume when calculating the sine, cosine and tangent.
With the frequent use of this table you will memorize the values. If you have difficulty memorizing, you can learn the song below or in the next topic find out how to find the values of the remarkable angles through mathematical calculations.
Song remarkable angles
Obtaining the values of the notable angles by means of mathematical calculations
To demonstrate how to obtain the values of sine, cosine and tangent for notable angles, sketch an equilateral triangle initially. Remember that: the equilateral triangle has all sides with the same measurement and all angles measure 60 °.
Next, we will determine the height of this triangle, to do this, plot the angle bisector (A). This bisector will meet the straight line (CB). The bisector will be the median, and the median will determine the midpoint of the line segment (CB).
We then have to:
Obs. The equilateral triangle has all the specific properties and characteristics that allow the height, bisector and median to be the same segment. It is important to note that there are many other cases in geometry where this does not occur.
We will now determine the height of this triangle by applying the Pythagorean Theorem on the ACD triangle, follow:
To obtain the values for the notable angles, consider only one side of the triangle represented above.
We will now apply the sine, cosine and tangent formulas already presented above.
After discovering the numerical values of sine, cosine and tangent for the notable angles 30 ° and 60 °, we still need to find out for 45 °. To obtain the values for the sine, cosine and tangent of this angle, we need to draw a square and draw its diagonal, see:
A square has all four internal angles measuring 90 °. When we draw the diagonal (d) of the square, we divide the angle of 90 ° in half, that is, the new angles become 45 °.
We will apply the Pythagorean Theorem to find the value of the diagonal of the triangle ABC in terms of (a).
With the value of the diagonal / hypotenuse and the collectors in terms of (a), we were able to calculate the sine, cosine and tangent of 45 °, follow:
If you are unable to memorize the values corresponding to the remarkable angles, now at least you know how to calculate them.
How to know when to use sine cosine and tangent
The image illustrates the use of the triangle on one of the most famous bridges in the world, the Golden Gate, in the United States (Photo: depositphotos)
We will use sine, cosine and tangent when we need to find the measure for either side of the right triangle or when we need to know the measure of the acute internal angles.
The structure of triangles is widely used in the construction of objects and structures , and can be easily found in civil construction. This is because the triangle is considered a rigid geometric figure, that is, it does not deform easily. Thus, any construction that has triangles in its structure is a more stable construction.
Remember that through geometric concepts it is possible to obtain right triangles in any other triangle.
Knowing how to use sine, cosine and tangent can help you if you ever need to build or model something and choose the geometric model of the triangle . You will know how to find the measure of the angles and sides of that triangle.
