Property 4
Definition of the Trigonometric Functions (Right Triangle)
c is the hypotenuse; b is opposite α; a is adjacent to α.
\(\cos\alpha=\samefrac{\text{Length of adjacent}}{\text{Length of hypotenuse}}=\samefrac ac\)
\(\sin\alpha=\samefrac{\text{Length of opposite}}{\text{Length of hypotenuse}}=\samefrac bc\)
\(\tan\alpha=\samefrac{\text{Length of opposite}}{\text{Length of adjacent}}=\samefrac ba\)
\(\cot\alpha=\samefrac{\text{Length of adjacent}}{\text{Length of opposite}}=\samefrac ab\)
Example
\(\sin\alpha=\samefrac35\)
\(\cos\alpha=\samefrac45\)
\(\tan\alpha=\samefrac34\)
\(\cot\alpha=\samefrac43\)
Find the trigonometric values of the angles of the right triangles below.
1.
\(\sin\alpha=?\)
\(\cos\alpha=?\)
\(\tan\alpha=?\)
\(\cot\alpha=?\)
2.
\(\sin\alpha=?\)
\(\cos\alpha=?\)
\(\tan\alpha=?\)
\(\cot\alpha=?\)
3.
\(\sin\alpha=?\)
\(\cos\alpha=?\)
\(\tan\alpha=?\)
\(\cot\alpha=?\)
4.
\(\sin\alpha=?\)
\(\cos\alpha=?\)
\(\tan\alpha=?\)
\(\cot\alpha=?\)
\(\sin\beta=?\)
\(\cos\beta=?\)
\(\tan\beta=?\)
\(\cot\beta=?\)
Property 5
If one trigonometric value of an acute angle is given, other trigonometric values can be found by drawing a right triangle.
Example
\(0<\alpha<90^\circ,\ \tan\alpha=3\)
\(\sin\alpha=\samefrac3{\sqrt{10}},\ \cos\alpha=\samefrac1{\sqrt{10}},\ \cot\alpha=\samefrac13\)
1.
\(0<x<90^\circ\)
\(\tan x=\samefrac12\)
\(\sin x=?\)
2.
\(0<x<90^\circ\)
\(\sin x=\samefrac45\)
\(\tan x=?\)
3.
\(0<x<90^\circ\)
\(\sin x=\samefrac5{13}\)
\(\cot x=?\)
4.
\(0<x<90^\circ\)
\(\tan x=\samefrac13\)
\(\cos x+\sin x=?\)
5.
ABCD square
\(\Rightarrow \tan\alpha=?\)
6.
The figure consists of congruent squares.
\(\Rightarrow \sin\alpha=?\)
7.
\(\text{Perimeter}(ABC)=24\)
\(\sin\alpha=\samefrac45\)
\(\Rightarrow x=?\)
8.
ABCD square
\(\Rightarrow \tan\alpha=?\)
9.
\(3\tan\widehat B=2\tan\widehat C\)
\(\Rightarrow |AB|=?\)