| 1. Trigonometric Functions
Trigonometric functions are also known as circular functions. Trigonometric functions describe the relationship between the angles and the sides of a triangle.
The six trigonometric functions are sin \(sin \text{ }\theta\) , cos \(cos \text{ }\theta\) , tan \(tan \text{ }\theta\) , csc \(csc \text{ }\theta\) , sec \(sec \text{ }\theta\) and cot \(cot \text{ }\theta\) whereby csc \(csc \text{ }\theta\) = \({1 \over sin \text{ } sin \text { } \theta}\) , sec \(sec \text { } \theta\) = \({1 \over cos \text{ } cos \text { } \theta}\) and cot \(cot \text { } \theta\) = \({1 \over tan \text{ } tan \text { } \theta}\) .
On a Cartesian plane, sin \(sin \text { } \theta\) is positive on the I and II quadrants, cos \(cos \text { } \theta\) is positive on the I and IV quadrants and tan \(tan \text { } \theta\) is positive on the I and III quadrants.
Their reciprocals functions have positive signs in the same quadrants. Thus, csc \(csc \text { } \theta\) , is positive on the I and II quadrants, sec \(sec \text { } \theta\) is positive on the I and IV quadrants and cot \(cot \text { } \theta\) is positive on the I and III quadrants.
In other quadrants, the trigonometric functions have negative signs.
2. Trigonometric Ratios
Consider a right triangle with base x, height y and hypotenuse r. The ratios for the six trigonometric functions are:
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\(sin \text{ } sin \text { } \theta = {opposite \over hypotenuse} = {y \over r}\)
\(cos \text{ } cos \text { } \theta = {adjecent \over hypotenuse} = {x \over r}\)
\(tan \text{ } tan \text { } \theta = {opposite \over adjacent} = {y \over x}\)
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Following that,
\(csc \text{ } csc\text { } \theta = {r \over y}\), \(sec \text{ } sec \text { } \theta ={r \over x}\), \(cot \text{ } cot \text { } \theta = {x \over y}\)
Take note that \(r^2=x^2+y^2.\)
Example
Example
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sin \(sin \text { } \theta\) =-2425 (because sin is negative in Quadrant III)
cos \(cos \text { } \theta\) =-725 (because cos is negative in Quadrant III)
tan \(tan \text { } \theta\) =247 (because tan is positive in Quadrant III)
csc \(csc \text { } \theta\) =-2524 (because csc is negative in Quadrant III)
sec \(sec \text { } \theta\) =-257 (because sec is negative in Quadrant III)
cot \(cot \text { } \theta\) =724 (because cot is positive in Quadrant III)
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Tag
Secondary school
Trigonometric functions
Trigonometric ratios
Reflection
What are the six trigonometric functions?
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How do you read the trigonometric ratios using the sides of a right triangle?
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Given the above right triangle in the fourth quadrant, what is the value of:
1. Hypotenuse?
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2. sin \(sin \text { } \theta\) ?
3. cos \(cos \text { } \theta\)?
4. tan \(tan \text { } \theta\)
5. csc \(csc \text { } \theta\)?
6. sec \(sec \text { } \theta\) ?
8. cot \(cot \text { } \theta\)?
What is the relationship between sin sin (theta), cos cos (theta), tan tan (theta) and csc csc (theta), sec sec (theta) and cot cot (theta) ?
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How do you read the six trigonometric ratios from a right triangle?
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