Question

The point (1, −1) is on the terminal side of angle θ, in standard position. What are the values of sine, cosine, and tangent of θ? Make sure to show all work.

Answer 1

The point (1, -1) allows us to find the values of sine, cosine, and tangent of θ by considering a right triangle and using trigonometric ratios. The calculated values are: cos(θ) = √2/2, sin(θ) = -√2/2, and tan(θ) = -1.

Step-by-step explanation:

The point (1, −1) is on the terminal side of angle θ, in standard position. To find the sine, cosine, and tangent of θ, we can consider a right triangle formed by the x-axis, the y-axis, and the terminal side of θ. The x-coordinate represents the length of the side adjacent to θ, the y-coordinate represents the length of the side opposite to θ, and the hypotenuse can be found using the Pythagorean theorem.

Let's calculate the hypotenuse (h) from the given coordinates (x, y):

h = √(x² + y²) = √(1² + (−1)²) = √(1 + 1) = √2

Therefore, the cosine (θ) is:

cos(θ) = x/h = 1/√2 = √2/2

And the sine (θ) is:

sin(θ) = y/h = (−1)/√2 = −√2/2

Finally, the tangent (θ) is simply the ratio of the opposite over the adjacent side, which gives us:

tan(θ) = y/x = (−1)/1 = −1

So, the values are cos(θ) = √2/2, sin(θ) = −√2/2, and tan(θ) = −1.

Answer 2

notice the picture below

x = 1 and y = -1

now... recall your SOH CAH TOA
`\bf sin(\theta)=\cfrac{y}{r} \qquad % cosine cos(\theta)=\cfrac{x}{r} \qquad % tangent tan(\theta)=\cfrac{y}{x}`

now.. what's the value for "r" or the radius? well, using the pythagorean theorem
`\bf c^2=a^2+b^2\implies c=√(a^2+b^2)\qquad \begin{cases} a=x\\ b=y\\ c=r \end{cases}`