Degree
An angle of a complete circumference is 360°. A radian is an angle whose arc is equal to the radius, expressed as S=d&r! 2=360°. Degrees and sexagesimal 0; 30; 45; 60; 90; 120; 90; 120; 135; 150; 180; 210; 225; 240; 270; 300; 315; 330; 360; and radians 0 TT/6 T/4T/3 m/2 2/3 3π/45π/6 π 7π/65m/44/33m/25m/37m/4.
Trigonometric ratios of an acute angle
Trigonometric ratios include the opposite over the hypotenuse equals sine, the adjacent over the hypotenuse equals cosine, and the opposite over the adjacent equals tangent. More trigonometric ratios include cosecant equals one over sine, secant equals one over cosine, cotangent equals one over tangent. Furthermore, the Pythagorean identity states that sine squared plus cosine squared of an angle equals 1.
Relationships between trigonometric ratios
There are different relationships between trigonometric ratios, including sine, cosine, adjacent, and hypotenuse. For instance, 1 plus tangent squared of an angle equals secant squared of that angle.
Trigonometric ratios of 45°, 60°, 30°
When the angle is 45°, the sine of 45° equals the square root of 2 over 2, the cosine of 45° is 1 over the square root of 2, and the tangent of 45° equals 1. When the angle is 60°, the sine of 60° is the square root of 3 over 2, the cosine of 60° equals 1 over 2, and the tangent of 60° equals the square root of 3.
Applications of trigonometry
Trigonometry is widely used for resolving right-angled triangles and calculating areas. It is also utilized in calculating double tangent problems and applying trigonometric functions to problem-solving.
Circumference goniometric and quadrants
In the circumference goniometric, if an angle is bigger than 0, it falls in the second quadrant. Sign changes of trigonometric ratios occur in different quadrants, for example, in the second quadrant, cosine is negative, sine is positive, and tangent is negative.
Reduction of angles to the first quadrant
Angles can be reduced to the first quadrant, an example is when two angles are supplementary, meaning they add up to 180°.
Trigonometric equations
Trigonometric equations involve trigonometric functions such as sine, cosine, and tangent. It is important to find solutions within a specified range, such as within 0° to 360°.
Double tangent exercise
An example of a double tangent exercise is solving for the hypotenuse and other sides of a triangle using the tangent function.
System of inequalities
Inequations are solved separately and then the common solution is identified. There are different types of inequalities, including linear and non-linear, and they are solved by finding the common solution of the individual inequalities.
Areas, volumes, and similarities
Geometry in the plane includes different formulas for different shapes, such as the area of a triangle, square, rectangle, rhombus, and regular polygon. In figures with circular shapes, formulas for circumference, sector, segment, crown, and circular trapezoid are utilized. In three-dimensional geometry, formulas for prisms and cubes are used to calculate volume and area.
This text covered various topics related to trigonometry including angles, trigonometric ratios, applications, reduction of angles, trigonometric equations, and geometrical shapes in the plane and in three-dimensional space. Calculating areas, volumes, and solving system of inequalities were also discussed. The use of trigonometry is essential in various fields, and understanding its fundamental principles is important for further advancement in mathematics and related disciplines. For additional resources and exercises to practice these concepts, a PDF file with solved exercises on trigonometric equations of the first and second degree, resolution of right-angled triangles, and reduction to the first quadrant can be found online.
