THE trigonometry establishes relationships between the measures of angles and segments. For such calculations, we use the trigonometric ratios that provide the values of the sine, cosine and tangentfrom acute angles. The best known and most used ratios are 30º, 45º and 60º, but the trigonometric tables present all the ratios involving the acute angles (< 90º).
In some situations involving distance calculations by measuring angles, there is a need to use obtuse angle ratios (> 90º). In these cases, we use formulas that relate the obtuse angles to the acute angles. Watch:
sin x = sin (180º - x)
The sine of an obtuse angle is equal to the sine of the supplement of that angle.
cos x = – cos (180º – x)
The cosine of an obtuse angle is the opposite of the cosine of the supplement of that angle.
Example 1
The 150º angle is obtuse, as its measurement value is greater than 90º. Let's determine the sine and cosine of this angle.
sin 150º = sin (180º - x)
sin 150º = sin (180º – 150º)
sin 150th = sin 30th
sin 30th = 1/2
Do not stop now... There's more after the advertising ;)
Then:
sin 150º = 1/2
cos 150º = -cos (180º - x)
cos 150º = -cos (180º - 150)
cos 150º = -cos 30º
–cos 30º = –√3/2
Thus:
cos 150º = –√3/2
Example 2
Determine the sine and cosine of 120º
sin 120° = sin (180° – 120°)
sin 120º = sin 60º
sin 60º = √3/2
then:
sin 120º = √3/2
cos 120º = –cos (180º – 120º)
cos 120º = -cos 60º
–cos 60º = – 1/2
then:
cos 120º = –1/2
Example 3
Determine the value of x in the following expressions:
x = sin 40º - sin 140º + cos 20º + cos 160º
sin 140° = sin (180° – 140°)
sin 140º = sin 40º
cos 160º = – cos (180º – 160º)
cos 160º = - cos 20º
x = sin 40º - sin 140º + cos 20º + cos 160º
x = sin 40º – sin 40º + cos 20º – cos 20º
x = 0
by Mark Noah
Graduated in Mathematics
Trigonometry - Math - Brazil School
Would you like to reference this text in a school or academic work? Look:
SILVA, Marcos Noé Pedro da. "Sine and Cosine of Obtuse Angles"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/seno-cosseno-Angulos-obtusos.htm. Accessed on June 27, 2021.
