Question

task 2 - knowledge: complete the following practice questions

1. find the value of each trig ratio. leave as an exact answer.

a) \\( \tan ( a ) \\)
b) \\( \sin ( x ) \\)
c) \\( \cos ( c ) \\)

1. use a calculator to find the value of each. round to 2 decimal places if necessary.

a) \\( \sin ( 20 ^ { \circ } ) \\)
b) \\( \cos ( 30 ^ { \circ } ) \\)
c) \\( \tan ( 40 ^ { \circ } ) \\)

1. solve for the indicated side length in each triangle. round to 2 decimal places if necessary.

a)
b)
c)
d)

Explanation:

1a)

Step1: Recall the tangent ratio formula

In a right - triangle, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). For \(\tan(A)\), the side opposite to angle \(A\) is \(BC = 30\) and the side adjacent to angle \(A\) is \(AB=16\).
\(\tan(A)=\frac{BC}{AB}\)

Step2: Substitute the values

Substitute \(BC = 30\) and \(AB = 16\) into the formula.
\(\tan(A)=\frac{30}{16}=\frac{15}{8}\)

1b)

Step1: Recall the sine ratio formula

In a right - triangle, \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). First, find the side opposite to angle \(X\). Using the Pythagorean theorem in \(\triangle XYZ\) (where \(XY = 12\), \(YZ = 16\), and \(XZ = 20\)), for \(\sin(X)\), the side opposite to angle \(X\) is \(YZ = 16\) and the hypotenuse is \(XZ = 20\).
\(\sin(X)=\frac{YZ}{XZ}\)

Step2: Substitute the values

Substitute \(YZ = 16\) and \(XZ = 20\) into the formula.
\(\sin(X)=\frac{16}{20}=\frac{4}{5}\)

1c)

Step1: Recall the cosine ratio formula

In a right - triangle, \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\cos(C)\), the side adjacent to angle \(C\) is \(BC = 27\) and the hypotenuse is \(AC = 45\).
\(\cos(C)=\frac{BC}{AC}\)

Step2: Substitute the values

Substitute \(BC = 27\) and \(AC = 45\) into the formula.
\(\cos(C)=\frac{27}{45}=\frac{3}{5}\)

2a)

Step1: Use a calculator

Set the calculator to degree mode. Then, calculate \(\sin(20^{\circ})\).
\(\sin(20^{\circ})\approx0.34\)

2b)

Step1: Use a calculator

Set the calculator to degree mode. Then, calculate \(\cos(30^{\circ})\).
\(\cos(30^{\circ})\approx0.87\)

2c)

Step1: Use a calculator

Set the calculator to degree mode. Then, calculate \(\tan(40^{\circ})\).
\(\tan(40^{\circ})\approx0.84\)

3a)

Step1: Recall the tangent ratio

In the right - triangle, \(\tan(27^{\circ})=\frac{x}{10}\).

Step2: Solve for \(x\)

\(x = 10\times\tan(27^{\circ})\). Using a calculator in degree mode, \(\tan(27^{\circ})\approx0.51\), so \(x\approx10\times0.51 = 5.10\)

3b)

Step1: Recall the sine ratio

In the right - triangle, \(\sin(54^{\circ})=\frac{x}{12}\).

Step2: Solve for \(x\)

\(x = 12\times\sin(54^{\circ})\). Using a calculator in degree mode, \(\sin(54^{\circ})\approx0.81\), so \(x\approx12\times0.81=9.72\)

3c)

Step1: Recall the tangent ratio

In the right - triangle, \(\tan(38^{\circ})=\frac{20}{x}\).

Step2: Solve for \(x\)

\(x=\frac{20}{\tan(38^{\circ})}\). Using a calculator in degree mode, \(\tan(38^{\circ})\approx0.78\), so \(x=\frac{20}{0.78}\approx25.64\)

3d)

Step1: Recall the cosine ratio

In the right - triangle, \(\cos(38^{\circ})=\frac{x}{20}\).

Step2: Solve for \(x\)

\(x = 20\times\cos(38^{\circ})\). Using a calculator in degree mode, \(\cos(38^{\circ})\approx0.79\), so \(x\approx20\times0.79 = 15.80\)

Answer:

1a) \(\frac{15}{8}\)
1b) \(\frac{4}{5}\)
1c) \(\frac{3}{5}\)
2a) \(0.34\)
2b) \(0.87\)
2c) \(0.84\)
3a) \(5.10\)
3b) \(9.72\)
3c) \(25.64\)
3d) \(15.80\)