solve this question pls don't spam answer is
I want full process. spammers would be reported
Answers
Answer:
the trigonometric functions (also called circular functions, angle functions or goniometric functions[1][2]) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.
Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of five acute angles.
The most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.[3]
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extending these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) is often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane (from which some isolated points are removed).In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; the same lower case letter denotes an edge of the triangle and its length.
Given an acute angle A = θ of a right-angled triangle, the hypotenuse h is the side that connects the two acute angles. The side b adjacent to θ is the side of the triangle that connects θ to the right angle. The third side a is said to be opposite to θ.
If the angle θ is given, then all sides of the right-angled triangle are well-defined up to a scaling factor. This means that the ratio of any two side lengths depends only on θ. Thus these six ratios define six functions of θ, which are the trigonometric functions. More precisely, the six trigonometric functions are:[4][5]
sine
{\displaystyle \sin \theta ={\frac {a}{h}}={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}{\displaystyle \sin \theta ={\frac {a}{h}}={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}
cosine
{\displaystyle \cos \theta ={\frac {b}{h}}={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}{\displaystyle \cos \theta ={\frac {b}{h}}={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}
tangent
{\displaystyle \tan \theta ={\frac {a}{b}}={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}{\displaystyle \tan \theta ={\frac {a}{b}}={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}
cotangent
{\displaystyle \cot \theta ={\frac {b}{a}}={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}{\displaystyle \cot \theta ={\frac {b}{a}}={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or {\textstyle {\frac {\pi }{2}}}{\textstyle {\frac {\pi }{2}}} radians.
Summary of relationships between trigonometric functions[6]
Function Abbreviation Description Relationship
using radians using degrees
sine sin
opposite
/
hypotenuse
{\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}\sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }} {\displaystyle \sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}}
cosine cos
adjacent
/
hypotenuse
{\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}\cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\, {\displaystyle \cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,}
tangent tan (or tg)
opposite
/
adjacent
{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}\tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }} {\displaystyle \tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}}
cotangent cot (or cotan or cotg or ctg or ctn)
adjacent
/