If cot A = 5/6, find cos A + csc A. Given cos θ = 1/3 and -π/2 ≤ θ ≤ 2π, find -8tan²θ - 16. If f(x) = sin 2x + cos 2x + sec 2x + csc 2x + tan 2x, what is f(π/3)? If terminal point... If cot A = 5/6, find cos A + csc A. Given cos θ = 1/3 and -π/2 ≤ θ ≤ 2π, find -8tan²θ - 16. If f(x) = sin 2x + cos 2x + sec 2x + csc 2x + tan 2x, what is f(π/3)? If terminal point of an arc of length θ lies on the line joining the origin and the point (-3,-1), what is cos²θ - sin²θ?
Understand the Problem
The question is asking for solutions to various trigonometric equations and identities. It includes finding specific values or results based on the provided trigonometric statements or functions.
Answer
$$ \frac{455 - 72\sqrt{61}}{30\sqrt{61}} $$
Answer for screen readers
The final answer is $$ \frac{455 - 72\sqrt{61}}{30\sqrt{61}} $$
Steps to Solve
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Calculate Values from Given Information
For ( \cot A = \frac{5}{6} ), we can find ( \sin A ) and ( \cos A ).
Using the identity ( \cot A = \frac{\cos A}{\sin A} ), we write $$ \cos A = 5k \quad \text{and} \quad \sin A = 6k $$ where ( k ) is a common factor. By the Pythagorean theorem, $$ \cos^2 A + \sin^2 A = 1 $$
Substituting gives: $$ (5k)^2 + (6k)^2 = 1 \implies 25k^2 + 36k^2 = 1 \implies 61k^2 = 1 \implies k^2 = \frac{1}{61} \implies k = \frac{1}{\sqrt{61}} $$
Hence, $$ \cos A = \frac{5}{\sqrt{61}}, \quad \sin A = \frac{6}{\sqrt{61}} $$
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Use the Formulas for ( \cos(A+C) ) and ( \sin(A) )
We need ( \cos A + \csc A ) and ( 2 \sec A \sin A ). First, $$ \csc A = \frac{1}{\sin A} = \frac{\sqrt{61}}{6} $$ Therefore, $$ \cos A + \csc A = \frac{5}{\sqrt{61}} + \frac{\sqrt{61}}{6} $$
To combine, find a common denominator (use ( 6\sqrt{61} )): $$ = \frac{30 + 61}{6\sqrt{61}} = \frac{91}{6\sqrt{61}} $$
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Calculate ( 2 \sec A \sin A )
First, find ( \sec A ): $$ \sec A = \frac{1}{\cos A} = \frac{\sqrt{61}}{5} $$ Thus, $$ 2 \sec A \sin A = 2 \cdot \frac{\sqrt{61}}{5} \cdot \frac{6}{\sqrt{61}} = \frac{12}{5} $$
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Final Calculation of the Required Expression
Now substitute these into the required result: $$ \cos A + \csc A - 2 \sec A \sin A = \frac{91}{6\sqrt{61}} - \frac{12}{5} $$
To combine, use the least common multiple for denominators: $$ = \frac{91 \cdot 5 - 12 \cdot 6\sqrt{61}}{30\sqrt{61}} $$ Calculating this gives: $$ \frac{455 - 72\sqrt{61}}{30\sqrt{61}} $$
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Conclusion
The final result is: $$ \frac{455 - 72\sqrt{61}}{30\sqrt{61}} $$
The final answer is $$ \frac{455 - 72\sqrt{61}}{30\sqrt{61}} $$
More Information
This is a multi-step problem involving trigonometric identities and simplifications. The solution involves using fundamental trigonometric identities and the Pythagorean theorem to derive necessary values.
Tips
- Forgetting to apply the Pythagorean theorem correctly to find trigonometric values.
- Not converting fractions to a common denominator when combining terms.
- Overlooking the negative signs when moving terms across the equation.
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