Question

Find the exact values of the six trigonometric functions of the angle θ shown in the
figure.

Answer 1

`\sf \sin(\theta)=(√(2))/(2)`

`\sf \cos(\theta)=(√(2))/(2)`

`\sf \tan(\theta)=1`

`\sf \csc(\theta)=√(2)`

`\sf \sec(\theta)=√(2)`

`\sf \cot(\theta)=1`

Explanation:

`\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}`

From inspection of the given right triangle:

• 
  `a = \sf 8`
• 
  `c = \sf 8√(2)`

Substitute the values of a and c into Pythagoras Theorem and solve for b:

`\implies 8^2+b^2=(8√(2))^2`

`\implies 64+b^2=128`

`\implies b^2=64`

`\implies b=8`

Therefore, the unknown side length opposite angle θ is 8 units.

`\boxed{\begin{minipage}{7 cm}\underline{Trigonometric ratios} \\\\$\sf \sin(\theta)=(O)/(H)\quad\cos(\theta)=(A)/(H)\quad\tan(\theta)=(O)/(A)$\\\\\\$\sf \csc(\theta)=(H)/(O)\quad\sec(\theta)=(H)/(A)\quad\cot(\theta)=(A)/(O)$\\\\where:\\\phantom{ww}$\bullet$ $\theta$ is the angle\\\phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle\\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle\\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse\\\end{minipage}}`

From inspection of the given right triangle:

• 
  `\sf O = 8`

• 
  `\sf A = 8`

• 
  `\sf H = 8 √(2)`

Substitute the values of O, A and H into the trigonometric ratios:

`\sf \sin(\theta)=(8)/(8√(2))=(1)/(√(2))=(√(2))/(2)`

`\sf \cos(\theta)=(8)/(8√(2))=(1)/(√(2))=(√(2))/(2)`

`\sf \tan(\theta)=(8)/(8)=1`

`\sf \csc(\theta)=(8√(2))/(8)=√(2)`

`\sf \sec(\theta)=(8√(2))/(8)=√(2)`

`\sf \cot(\theta)=(8)/(8)=1`