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Trigonometric Ratios for Complementary and Supplementary Angles

sin (–θ) = –sin θ cos(–θ) = cos θ
sin (90° – θ) = cos θ cos(90° – θ) = sin θ
sin(90° + θ) = cos θ cos(90° + θ) = –sin θ
sin(180° – θ) = sin θ cos(180° – θ) = –cos θ
sin(180° + θ) = –sin θ cos(180° + θ) = –cos θ
 
Since the terminal sides of co-terminal angles coincide, hence their trigonometrical ratios are same.
 
Clearly, 360° – θ and –θ are co-terminal angles. Therefore, sin(360° – θ) = sin(–θ) = –sin θ, cos(360° – θ) = cos(–θ) = cos θ, and tan(360° – θ) = tan(–θ) = –tan θ.
 
Similarly, cosec(360° – θ) = –cosec θ, sec(360° – θ) = sec θ and cot(360° – θ) = –cot θ.
 
We know that the terminal sides of co–terminal angles always coincide and θ and 360° + θ are co-terminal angles. Therefore, sin(360° + θ) = sin θ, cos(360° + θ) = cos θ, tan(360° + θ) = tan θ, sec(360° + θ) = sec θ, cosec(360° + θ) = cosec θ, and cot(360° + θ) = cot θ.
 
In fact, for any positive integer n, (360° × n + θ) is co-terminal to θ. Therefore, for any positive integer n, we have
 
sin(360° × n + θ) = sin θ, cos (360° × n + θ) = cos θ
 
tan(360° × n + θ) = tan θ, cosec(360° × n + θ) = cosec θ
 
sec(360° × n + θ) = secθ, cot(360° × n + θ) = cot θ
 
cos (nπ + θ) = (–1)n cos θn  I and sin (nπ + θ) = (–1)n sin θn  I
 
cos105686.png sin θn is odd integer
 
sin105680.png n is odd integer




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