TOPICS - TRIGONOMETRY: SINE, COS, TAN - REVIEW

How to define sine, cosine, and tangent in a right triangle?
What are the relations between the sides of a right triangle and the trigonometric ratios?
How to apply the Pythagorean Theorem to find unknown sides in a right triangle?
How can we use trigonometric functions to solve practical problems?
How do angles affect the values of sine, cosine, and tangent?

Sine Definition: Ratio between the opposite cathetus and the hypotenuse.
Cosine Definition: Ratio between the adjacent cathetus and the hypotenuse.
Tangent Definition: Ratio between the opposite cathetus and the adjacent cathetus.
Application of trigonometric relations in everyday situations and geometric problems.

Sine (sin): $$ sin(\theta) = \frac{opposite\ cathetus}{hypotenuse} $$
Cosine (cos): $$ cos(\theta) = \frac{adjacent\ cathetus}{hypotenuse} $$
Tangent (tan): $$ tan(\theta) = \frac{opposite\ cathetus}{adjacent\ cathetus} $$
Pythagorean Theorem: $$ a^2 + b^2 = c^2 $$, where $$c$$ is the hypotenuse and $$a$$ and $$b$$ are the catheti of the right triangle.

Trigonometry: a branch of mathematics that studies the relationships between angles and sides of triangles.
Sine (sin): a trigonometric function that, in a right triangle, relates the acute angle to the ratio between the length of the opposite cathetus and the hypotenuse.
Cosine (cos): a trigonometric function that relates the acute angle to the ratio between the length of the adjacent cathetus and the hypotenuse.
Tangent (tan): a trigonometric function that relates the acute angle to the ratio between the length of the opposite cathetus and the adjacent cathetus.
Hypotenuse: the longest side of a right triangle, opposite the right angle.
Catheti: the two shorter sides of a right triangle, forming the right angle.

Trigonometry is essential in various areas such as engineering, physics, architecture, and navigation.
The ratios sine, cosine, and tangent are constant proportions for a given angle, regardless of the size of the triangle.
The Pythagorean Theorem is the basis for metric relations in right triangles and is essential to understand trigonometry.

The sine of an acute angle is a ratio that can be found in any right triangle containing the angle, always relating it to the opposite side and the hypotenuse.
Cosine follows a similar reasoning to sine but relates the angle to the adjacent side.
Tangent is unique as it relates two sides that are not the hypotenuse, that is, it is a ratio between catheti.

For a right triangle with sides of lengths 3, 4, and 5, where 5 is the hypotenuse:
- The sine of the angle opposite the side of length 3: $$ sin(\theta) = \frac{3}{5} $$
- The cosine of the angle opposite the side of length 4: $$ cos(\theta) = \frac{4}{5} $$
- The tangent of the angle formed by the sides of lengths 3 and 4: $$ tan(\theta) = \frac{3}{4} $$
Using the Pythagorean Theorem, we can verify that the triangle with sides 3, 4, 5 is right-angled: $$ 3^2 + 4^2 = 5^2 $$
These values are used to calculate the height of an object from a known distance, observing the angle of elevation to the top of the object.

Trigonometry focuses on the relationships between angles and sides of triangles, especially right triangles.
Sine, cosine, and tangent are ratios between the sides of a right triangle that correspond to specific acute angles.
The Pythagorean Theorem is essential for solving right triangles and understanding the relationship between their sides.

Sine, cosine, and tangent are essential tools for addressing geometric and practical problems.
Knowing an angle and a side allows calculating the other sides of a right triangle.
The relationship between trigonometric ratios and the sides of the triangle is constant, regardless of the size of the triangle.
Illustrating with a triangle of sides 3, 4, and 5: sin(θ) = 3/5, cos(θ) = 4/5, tan(θ) = 3/4 highlights the direct calculation of the ratios.
These concepts are applicable in different contexts, from physics to navigation, reinforcing the interdisciplinary nature of trigonometry.

Summary of Trigonometry: Sine, Cosine, and Tangent: Review