Question

find the values of the six trigonometric functions of an angle in standard position if the point with coordinates (13, 84) lies on its terminal side. sin α = 85/84, cos α = 85/13, tan α = 13/84, csc α = 84/85, sec α = 13/85, cot α = 84/13 sin α = 13/85, cos α = 84/85, tan α = 13/84, csc α = 85/13, sec α = 85/84, cot α = 84/13 sin α = 84/85, cos α = 13/85, tan α = 84/13, csc α = 85/84, sec α = 85/13, cot α = 13/84 sin α = 84/13, cos α = 84/85, tan α = 13/85, csc α = 13/84, sec α = 85/84, cot α = 85/13

Explanation:

Step1: Calculate the radius \(r\)

Given the point \((x = 13,y = 84)\) on the terminal - side of the angle. Use the formula \(r=\sqrt{x^{2}+y^{2}}\), so \(r=\sqrt{13^{2}+84^{2}}=\sqrt{169 + 7056}=\sqrt{7225}=85\).

Step2: Calculate sine function

By the definition \(\sin\alpha=\frac{y}{r}\), substituting \(y = 84\) and \(r = 85\), we get \(\sin\alpha=\frac{84}{85}\).

Step3: Calculate cosine function

By the definition \(\cos\alpha=\frac{x}{r}\), substituting \(x = 13\) and \(r = 85\), we get \(\cos\alpha=\frac{13}{85}\).

Step4: Calculate tangent function

By the definition \(\tan\alpha=\frac{y}{x}\), substituting \(x = 13\) and \(y = 84\), we get \(\tan\alpha=\frac{84}{13}\).

Step5: Calculate cosecant function

Since \(\csc\alpha=\frac{r}{y}\), substituting \(r = 85\) and \(y = 84\), we get \(\csc\alpha=\frac{85}{84}\).

Step6: Calculate secant function

Since \(\sec\alpha=\frac{r}{x}\), substituting \(r = 85\) and \(x = 13\), we get \(\sec\alpha=\frac{85}{13}\).

Step7: Calculate cotangent function

Since \(\cot\alpha=\frac{x}{y}\), substituting \(x = 13\) and \(y = 84\), we get \(\cot\alpha=\frac{13}{84}\).

Answer:

\(\sin\alpha=\frac{84}{85},\cos\alpha=\frac{13}{85},\tan\alpha=\frac{84}{13},\csc\alpha=\frac{85}{84},\sec\alpha=\frac{85}{13},\cot\alpha=\frac{13}{84}\) (corresponding to the third - option in the multiple - choice list)