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Math  /  Trigonometry

QuestionQuestion 6 - 5 marks Which is the correct ratio for csc⁡θ\csc \theta ? a) 513\frac{5}{13} b) 135\frac{13}{5} c) 1312\frac{13}{12} d) 125\frac{12}{5}

Studdy Solution

STEP 1

1. We are dealing with a right triangle.
2. The sides of the triangle are labeled as 5, 12, and 13.
3. The angle θ\theta is opposite the side labeled 5.
4. We need to find the correct ratio for csc⁡θ\csc \theta.

STEP 2

1. Understand the definition of csc⁡θ\csc \theta.
2. Identify the sides of the triangle relative to θ\theta.
3. Calculate csc⁡θ\csc \theta using the identified sides.
4. Match the calculated value with the given options.

STEP 3

The cosecant function, csc⁡θ\csc \theta, is defined as the reciprocal of the sine function. Thus, csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}.

STEP 4

In a right triangle, sin⁡θ=opposite sidehypotenuse\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}.
For angle θ\theta, the opposite side is 5, and the hypotenuse is 13.

STEP 5

Calculate sin⁡θ\sin \theta: sin⁡θ=513\sin \theta = \frac{5}{13}
Calculate csc⁡θ\csc \theta: csc⁡θ=1sin⁡θ=1513=135\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{5}{13}} = \frac{13}{5}

STEP 6

Compare the calculated csc⁡θ=135\csc \theta = \frac{13}{5} with the given options.
The correct option is: b) 135\frac{13}{5}
The correct ratio for csc⁡θ\csc \theta is:
135 \boxed{\frac{13}{5}}

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