Question

Let (4, -7) be a point on the terminal side of theta. Find the exact values of sin theta, secant theta, and tangent theta

Answer 1

To find the values of sine, secant, and tangent, we use the coordinates of the given point (4, -7). In a right triangle, sine is opposite/hypotenuse, secant is reciprocal of cosine, and tangent is opposite/adjacent.

Step-by-step explanation:

To find the exact values of sine, secant, and tangent, we need to use the coordinates of the given point (4, -7) on the terminal side of theta. In a right triangle, the sine is the ratio of the opposite side to the hypotenuse, the secant is the reciprocal of the cosine (which is the ratio of the adjacent side to the hypotenuse), and the tangent is the ratio of the opposite side to the adjacent side.

Given that the point (4, -7) is on the terminal side of theta, we can determine the lengths of the sides of the right triangle formed by this point and the origin (0, 0). The adjacent side is 4 units long, and the opposite side is -7 units long (since it is below the x-axis). The hypotenuse can be found using the Pythagorean theorem: hypotenuse^2 = adjacent^2 + opposite^2.

So, the hypotenuse^2 = 4^2 + (-7)^2 = 16 + 49 = 65. Taking the square root of both sides, we find that the hypotenuse is sqrt(65). Therefore, sin theta = opposite/hypotenuse = -7/sqrt(65), sec theta = 1/cos theta = 1/adjacent/hypotenuse = sqrt(65)/4, and tan theta = opposite/adjacent = -7/4.

Answer 2

Given:

The point on the terminal side of theta is (4,-7).

To find:

The exact values of sin theta, secant theta, and tangent theta.

Solution:

If a point
`P(x,y)` is on the terminal side of theta and
`r=√(x^2+y^2)`, then

`\sin \theta=(y)/(r)`

`\cos \theta=(x)/(r)`

`\sec \theta=(r)/(x)`

`\tan \theta=(y)/(x)`

The point on the terminal side of theta is (4,-7). Here,
`x=4` and
`y=-7`.

`r=√((4)^2+(-7)^2)`

`r=√(16+49)`

`r=√(65)`

Now,

`\sin \theta=(-7)/(√(65))`

`\cos \theta=(4)/(√(65))`

`\sec \theta=(√(65))/(4)`

`\tan \theta=(-7)/(4)`

Therefore,
`\sin \theta=(-7)/(√(65)),\cos \theta=(4)/(√(65)),\sec \theta=(√(65))/(4),\tan \theta=(-7)/(4)`.