Math Problem Statement

No it’s sin -5/13

Solution

Thank you for the correction! Let's solve the problem with the correct information that sinθ=513\sin \theta = -\frac{5}{13}.

Updated Problem Summary:

We are given that sinθ=513\sin \theta = -\frac{5}{13} with θ\theta in the fourth quadrant. We need to evaluate the expression:

secθ+26sinθ4tanθ\sec \theta + 26 \sin \theta - 4 \tan \theta

Step 1: Determine the other trigonometric values

Given: sinθ=513\sin \theta = -\frac{5}{13}

Since sinθ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, the opposite side is 5-5 and the hypotenuse is 1313. To find the adjacent side xx, we use the Pythagorean theorem:

x2+(5)2=132x^2 + (-5)^2 = 13^2 x2+25=169x^2 + 25 = 169 x2=144    x=12x^2 = 144 \implies x = 12

Since θ\theta is in the fourth quadrant, the adjacent side xx is positive. So, x=12x = 12.

Now, we can find the other trigonometric ratios: cosθ=adjacenthypotenuse=1213\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13} tanθ=oppositeadjacent=512\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{-5}{12}

Step 2: Evaluate the required expression

We now substitute these values into the expression:

  1. Secant: secθ=1cosθ=11213=1312\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{12}{13}} = \frac{13}{12}

  2. Sin: 26sinθ=26×(513)=1026 \sin \theta = 26 \times \left(-\frac{5}{13}\right) = -10

  3. Tangent: 4tanθ=4×(512)=2012=534 \tan \theta = 4 \times \left(\frac{-5}{12}\right) = -\frac{20}{12} = -\frac{5}{3}

Step 3: Combine all the terms

Now, let's add them together: secθ+26sinθ4tanθ=1312+(10)(53)\sec \theta + 26 \sin \theta - 4 \tan \theta = \frac{13}{12} + (-10) - \left(-\frac{5}{3}\right)

Convert 10-10 into a fraction with denominator 12: 10=12012-10 = -\frac{120}{12}

Now, the expression is: 131212012+2012=13120+2012=8712\frac{13}{12} - \frac{120}{12} + \frac{20}{12} = \frac{13 - 120 + 20}{12} = \frac{-87}{12}

Simplify the fraction: 8712=294\frac{-87}{12} = -\frac{29}{4}

Final Answer:

secθ+26sinθ4tanθ=294\sec \theta + 26 \sin \theta - 4 \tan \theta = -\frac{29}{4}

Would you like any additional explanations or have any further questions?

Here are some related questions that might interest you:

  1. Why is it important to determine the correct quadrant when evaluating trigonometric functions?
  2. How do you simplify trigonometric expressions involving multiple operations?
  3. What are the typical steps to verify if your trigonometric identities are correct?
  4. How can the Pythagorean theorem be applied in different contexts?
  5. Can you explain the relationship between sinθ\sin \theta and cosθ\cos \theta in different quadrants?

Tip: When dealing with trigonometric functions, always make sure to account for the signs based on the quadrant in which the angle lies.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities
Quadrants

Formulas

Pythagorean theorem

Theorems

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Suitable Grade Level

Grades 10-12