Find the sine, cosine, and tangent of each angle using the unit circle. θ = − 1.18 \theta=-1.18 θ = − 1.18 rad, ( 5 13 , − 12 13 ) \left(\frac{5}{13},-\frac{12}{13}\right) ( 13 5 , − 13 12 )

In this problem, we're asked to find the sine, cosine, and tangent of each angle using the unit circle. And here we have our given angle as negative 1.18 radians. So our value of theta, our angle is negative 1.18 radians. Now remember, whenever we're working with a negative angle, that just means that instead of going counterclockwise around my circle, I'm instead just going clockwise, so nothing to worry about here. Now let's go ahead and look at our unit circle and determine what exactly these trig values are. Now looking here, I have the point five over 13 and a negative 12 over 13. Now remember on the unit circle, this ordered pair will tell me my values of cosine and sine. So looking at this, five over 13 and negative 12 over 13, remember I have my x y values which correspond to my cosine thinking about that in alphabetical order. So here I have this five over 13 that I can fill in for my cosine. So the cosine of negative 1.18 radians is five over 13. Now my sine is my y value so don't forget that negative here. So my sine of negative 1.18 radians is going to be negative 12 over 13. Now a little bit more work with our tangent here but nothing too crazy because remember the tangent of our angle is going to be equal to the sine divided by the cosine or we can also think about this in terms of our ordered pairs y over x. So here I can take my y value which is negative 12 over 13 and divide it by my x value which is five over 13. Now this looks a little crazy a fraction divided by a fraction and you can think about this two different ways. So the first way you can think of is, okay, if I'm dividing by a fraction, I'm really just multiplying by the reciprocal of the denominator. But here, since these have the same denominators, we can also think of this as just dividing those numerators because those thirteens will end up canceling out. So this fraction is really negative 12 over five so the tangent of negative 1.18 radians is going to be negative 12 over five. Make sure you don't forget that negative. So that's all for this one. Thanks for watching and I'll see you in the next one.

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

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Multiple Choice

Find the sine, cosine, and tangent of each angle using the unit circle.
$\theta=-1.18$ rad, $\left(\frac{5}{13},-\frac{12}{13}\right)$

$\sin\theta=-\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=\frac{12}{5}$

$\sin\theta=-\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=-\frac{12}{5}$

$\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=\frac{5}{12}$

$\sin\theta=\frac{5}{13},\cos\theta=\frac{-12}{13},\tan\theta=\frac{5}{12}$

Verified step by step guidance

Step 1: Understand the unit circle representation. The unit circle is a circle with a radius of 1 centered at the origin (0,0). The coordinates of a point on the unit circle correspond to the cosine and sine of the angle θ, respectively.

Step 2: Identify the given angle and coordinates. The angle θ is -1.18 radians, and the coordinates of the point on the unit circle are (5/13, -12/13). This means cos(θ) = 5/13 and sin(θ) = -12/13.

Step 3: Recall the definition of tangent. The tangent of an angle θ is defined as tan(θ) = sin(θ) / cos(θ). Using the given values, tan(θ) = (-12/13) / (5/13). Simplify this fraction to find tan(θ).

Step 4: Verify the signs of the trigonometric functions. Since the angle is in the fourth quadrant (negative y-coordinate and positive x-coordinate), sin(θ) is negative, cos(θ) is positive, and tan(θ) (which is the ratio of sin to cos) is negative.

Step 5: Match the calculated values to the correct answer. Based on the calculations and quadrant analysis, the correct values are sin(θ) = -12/13, cos(θ) = 5/13, and tan(θ) = -12/5.