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PilotLPTM [1.2K]

3 years ago

9

Order the numbers from least to greatest: 82, 62, 95, 81, 89, 51, 72, 56, 97, 98, 79, 85

1 answer:

yarga [219]3 years ago

5 0

51,56,62,72,79,81,82,85,89,95,97,98

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3 years ago

The angle,2Θ, lies in the third quadrant such that cos2Θ=-2/5. Determine an exact value for tanΘ . Show your work including any

horrorfan [7]

Answer:

$tan(\theta)=\frac{\sqrt{21}}{3}$

Step-by-step explanation:

1. Approach

One is given the following information:

$cos(2\theta)=-\frac{2}{5}$

One can rewrite this as:

$cos(2\theta)=-0.4$

Also note, the problem says that the angle ( $2\theta$ ) is found in the third quadrant.

Using the trigonometric identities ( $cos(2\theta)=2(cos^2(\theta))-1$ ) and ( $cos(2\theta)=1-2(sin^2(\theta))$ ) one can solve for the values of ( $cos(\theta)$ ) and ( $sin(\theta)$ ). After doing so one can use another trigonometric identity ( $tan(\theta)=\frac{sin(\theta)}{cos(\theta)}$ ). Substitute the given information into the ratio and simplify.

2. Solve for $(cos(\theta))$

Use the following identity to solve for ( $cos(\theta)$ ) when given the value ( $cos(2\theta)$ ).

$cos(2\theta)=2(cos^2(\theta))-1$

Substitute the given information in and solve for ( $cos(\theta)$ ).

$cos(2\theta)=2(cos^2(\theta))-1$

$-0.4=2(cos^2(\theta))-1$

Inverse operations,

$-0.4=2(cos^2(\theta))-1$

$0.6=2(cos^2(\theta))$

$0.3=cos^2(\theta)$

$\sqrt{0.3}=cos(\theta)$

Since this angle is found in the third quadrant its value is actually:

$cos(\theta)=-\sqrt{0.3}$

3. Solve for $(sin(\theta))$

Use the other identity to solve for the value of ( $sin(\theta)$ ) when given the value of ( $cos(2\theta)$ ).

$cos(2\theta)=1-2(sin^2(\theta))$

Substitute the given information in and solve for ( $sin(\theta)$ ).

$cos(2\theta)=1-2(sin^2(\theta))$

$-0.4=1-2(sin^2(\theta))$

Inverse operations,

$-0.4=1-2(sin^2(\theta))$

$-1.4=-2(sin^2(\theta))$

$0.7=sin^2(\theta)$

$\sqrt{0.7}=sin(\theta)$

Since this angle is found in the third quadrant, its value is actually:

$sin(\theta)=-\sqrt{0.7}$

4. Solve for $(tan(\theta))$

One can use the following identity to solve for $(tan(\theta))$ ;

$tan(\theta)=\frac{sin(\theta)}{cos(\theta)}$

Substitute the values on just solved for and simplify,

$tan(\theta)=\frac{sin(\theta)}{cos(\theta)}$

$tan(\theta)=\frac{-\sqrt{0.7}}{-\sqrt{0.3}}$

$tan(\theta)=\frac{\sqrt{0.7}}{\sqrt{0.3}}$

$tan(\theta)=\frac{\sqrt{\frac{7}{10}}}{\sqrt{\frac{3}{10}}}$

Rationalize the denominator,

$tan(\theta)=\frac{\sqrt{\frac{7}{10}}}{\sqrt{\frac{3}{10}}}$

$tan(\theta)=\frac{\sqrt{\frac{7}{10}}}{\sqrt{\frac{3}{10}}}*\frac{\sqrt{\frac{3}{`0}}}{\sqrt{\frac{3}{10}}}$

$tan(\theta)=\frac{\sqrt{\frac{7}{10}*\frac{3}{10}}}{\sqrt{\frac{3}{10}*\frac{3}{10}}}$

$tan(\theta)=\frac{\sqrt{\frac{21}{100}}}{\frac{3}{10}}$

$tan(\theta)=\frac{\frac{\sqrt{21}}{10}}{\frac{3}{10}}$

$tan(\theta)=\frac{\sqrt{21}}{10}*\frac{10}{3}$

$tan(\theta)=\frac{\sqrt{21}}{3}$

4 0

3 years ago