Question

(a) Evaluate sin(5/4) (b) Evaluate tan(167/3) (c) Evaluate sec(-7π/6).

Answer 1

(a) Evaluate
`\displaystyle \sin\left((5)/(4)\right)`:

To evaluate
`\displaystyle \sin\left((5)/(4)\right)`, we use the trigonometric function sine, which gives us the ratio of the length of the side opposite to the angle to the length of the hypotenuse in a right triangle.

Since
`\displaystyle (5)/(4)` is in radians, we can find the sine value by taking the sine of
`\displaystyle (5)/(4)`:

`\displaystyle \sin\left((5)/(4)\right) = \sin\left(1.25\pi\right)`.

The value of
`\displaystyle \sin\left((5)/(4)\right)` is approximately -0.707.

(b) Evaluate
`\displaystyle \tan\left((167)/(3)\right)`:

To evaluate
`\displaystyle \tan\left((167)/(3)\right)`, we use the trigonometric function tangent, which gives us the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle in a right triangle.

Since
`\displaystyle (167)/(3)` is in degrees, we can convert it to radians by multiplying it by
`\displaystyle (\pi)/(180)`:

`\displaystyle \tan\left((167)/(3)\right) = \tan\left((167)/(3)* (\pi)/(180)\right)`.

The value of
`\displaystyle \tan\left((167)/(3)\right)` is approximately 0.999.

(c) Evaluate
`\displaystyle \sec\left(-(7\pi)/(6)\right)`:

To evaluate
`\displaystyle \sec\left(-(7\pi)/(6)\right)`, we use the trigonometric function secant, which is the reciprocal of the cosine function.

Since
`\displaystyle -(7\pi)/(6)` is in radians, we can find the cosine value and take its reciprocal:

`\displaystyle \sec\left(-(7\pi)/(6)\right) = (1)/(\cos\left(-(7\pi)/(6)\right))`.

The value of
`\displaystyle \sec\left(-(7\pi)/(6)\right)` is approximately 2.0.

Therefore, the evaluations are:

(a)
`\displaystyle \sin\left((5)/(4)\right) \approx -0.707`.

(b)
`\displaystyle \tan\left((167)/(3)\right) \approx 0.999`.

(c)
`\displaystyle \sec\left(-(7\pi)/(6)\right) \approx 2.0`.

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