Reciprocal Trigonometric Functions
Define and evaluate reciprocal trig functions cosecant, secant, and cotangent: each is the reciprocal of a primary trig function and extends the Grade 10 trigonometry toolkit.
Key Concepts
The cosecant, secant, and cotangent are reciprocals of the sine, cosine, and tangent functions: $$\csc A = \frac{1}{\sin A} = \frac{\text{hyp}}{\text{opp}}$$ $$\sec A = \frac{1}{\cos A} = \frac{\text{hyp}}{\text{adj}}$$ $$\cot A = \frac{1}{\tan A} = \frac{\text{adj}}{\text{opp}}$$.
If $\sin A = \frac{8}{17}$, then its reciprocal is $\csc A = \frac{17}{8}$. If $\cos A = \frac{15}{17}$, then its reciprocal is $\sec A = \frac{17}{15}$. If $\tan A = \frac{8}{15}$, then its reciprocal is $\cot A = \frac{15}{8}$.
These three are just the main trig functions flipped upside down! If you know SOH CAH TOA, you know these. Just flip the fraction of sine, cosine, or tangent to find them.
Common Questions
What are the three reciprocal trigonometric functions?
Cosecant csc(theta)=1/sin(theta), Secant sec(theta)=1/cos(theta), and Cotangent cot(theta)=1/tan(theta)=cos(theta)/sin(theta). Each is defined as the reciprocal of a primary trig function and is undefined when the denominator function equals zero.
How do you evaluate csc(30 degrees) using the primary trig functions?
Find sin(30)=1/2, then take the reciprocal: csc(30)=1/(1/2)=2. The same approach applies to all reciprocal functions: evaluate the primary function first, then flip the fraction.
When are reciprocal trig functions used in Grade 10 algebra?
Reciprocal functions appear in identities such as the Pythagorean identity 1+cot^2(theta)=csc^2(theta), in simplifying trig expressions, and in advanced math. In Saxon Algebra 2 they complete the six-function framework used throughout trigonometry.