Question

Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7.sin θ = √5/7 , and θ is in quadrant I.

Accepted Answer

Recall the six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). We are given sin $$ 	heta = \frac{\sqrt{5}}{7} $$ and that $$ 	heta  is in $$quadrant I, where all trigonometric functions are positive. Use the Pythagorean identity to find cos $$ 	heta $$: $$ \sin^2 	heta + \cos^2 	heta = 1 . $$Substitute $$ \sin 	heta = \frac{\sqrt{5}}{7}  to $$get $$ \left( \frac{\sqrt{5}}{7} ight)^2 + \cos^2 	heta = 1 . $$Simplify the equation: $$ \frac{5}{49} + \cos^2 	heta = 1 . $$Then solve for $$ \cos^2 	heta  by $$subtracting $$ \frac{5}{49} $$ from both sides: $$ \cos^2 	heta = 1 - \frac{5}{49} . $$Calculate $$ \cos 	heta  by $$taking the positive square root (since $$ 	heta  is in $$quadrant I): $$ \cos 	heta = \sqrt{1 - \frac{5}{49}} . $$Find the remaining trigonometric functions using the definitions: $$ 	an 	heta = \frac{\sin 	heta}{\cos 	heta} $$, $$ \csc 	heta = \frac{1}{\sin 	heta} $$, $$ \sec 	heta = \frac{1}{\cos 	heta} $$, and $$ \cot 	heta = \frac{1}{	an 	heta} . $$Rationalize denominators where necessary.

Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7.
sin θ = √5/7 , and θ is in quadrant I.

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주요 개념

Definition of the Six Trigonometric Functions

Using the Pythagorean Identity to Find Cosine

Quadrant Sign Rules for Trigonometric Functions