Trigonometric functions of the acute angle of a rectangular triangle. The sine theorem. The cosine theorem

The sine, cosine, tangent, and cotangent of an acute angle of a rectangular triangle

The opposite leg for ∠A is the leg BC, the adjacent leg is the leg AC. Accordingly, for ∠B the opposite leg is the leg AC, and the adjacent leg is the leg BC.
1. The sine of an acute angle of a rectangular triangle is the ratio of the opposite leg to the hypotenuse: sin∠A=BC/AB.
2. The cosine of an acute angle of a rectangular triangle is the ratio of the adjacent leg to the hypotenuse: cos∠A=AC/AB.
3. The tangent of an acute angle of a rectangular triangle is the ratio of the opposite leg to the adjacent leg: tg∠A=BC/AC.
4. The cotangent of an acute angle of a rectangular triangle is the ratio of the adjacent leg to the opposite leg: ctg∠A=AC/BC.
Sine, cosine, tangent, cotangent of an angle depend only on the size of the angle and do not depend on the size and location of the triangle.
Basic trigonometric formulas:

Values of trigonometric functions of angles 30°, 45°, 60°.

1. The angle 30°:

sin30° = a/2a = 1/2;    cos30° = √3/2;    tg30° = 1/√3;    ctg30° = √3.

2. The angle 60°:

sin60° = √3/2;    cos60° = √1/2;    tg60° = √3;    ctg60° = 1/√3.

3. The angle 45°:

sin45° = √2/2;    cos45° = √2/2;    tg45° = 1;    ctg45° = 1.

The sine theorem

The sides of a triangle are proportional to the sines of the opposite angles. The proportionality coefficient is 2R, where R is the radius of the circumscribed circle:

a/sinα = b/sinβ = c/sinγ = 2R

The cosine theorem

The square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them:

a² = b² + c² - 2bc cosα