Identify what angle, θ \theta θ , satisfies the following conditions. tan θ = − 1 \tan\theta=-1 tan θ = − 1 ; cos θ \cos\theta cos θ > 0 0 0

In this problem we're asked to identify which angle theta satisfies the following conditions and the conditions that we're given here are that the tangent of theta is equal to negative one and the cosine of theta is greater than zero. Now Now we want to find an angle that satisfies both of these conditions, but let's first take a look at this first condition, the tangent of theta being equal to negative one. Now looking at my unit circle here, I look at my tangent values and I see that for 135 degrees my tangent value is indeed negative one. Now continuing around my unit circle and ending up in quadrant four, I see that for three fifteen degrees my tangent is also equal to negative one. So there are two possible answers here either 135 degrees or three fifteen degrees. This is where our second condition comes in. Now our second condition here is that the cosine of theta has to be positive or greater than zero. Now in my second quadrant, quadrant two, my cosine value, this first value is negative. So this cannot be my answer. My answer is not 135 degrees. Now taking a look at our other angle to make sure that it does satisfy that other condition before just saying that's our answer, I come down here to my cosine value of root two over two, which is a positive value greater than zero. And I remember that in this quadrant four, cosine is actually the only positive trig function. So this is my final answer. That three fifteen degrees the tangent is equal to negative one and the cosine is greater than zero. So my angle here theta is three fifteen degrees and I'm done here. 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

Struggling with Business Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Identify what angle, $\theta$ , satisfies the following conditions.
$\tan\theta=-1$ ; $\cos\theta$ > $0$

45°

135°

315°

330°

Verified step by step guidance

Step 1: Recall the definition of the tangent function. The tangent of an angle, tan(θ), is defined as the ratio of the sine of the angle to the cosine of the angle: tan(θ) = sin(θ) / cos(θ).

Step 2: Analyze the given condition tan(θ) = -1. This means that the sine and cosine of the angle must have equal magnitudes but opposite signs (one positive and one negative).

Step 3: Consider the condition cos(θ) > 0. This tells us that the cosine of the angle is positive, which restricts θ to either the first quadrant (0° to 90°) or the fourth quadrant (270° to 360°), as cosine is positive in these quadrants.

Step 4: Identify the quadrants where tan(θ) = -1. Tangent is negative in the second quadrant (90° to 180°) and the fourth quadrant (270° to 360°). Since cos(θ) > 0, we focus on the fourth quadrant.

Step 5: Determine the specific angle in the fourth quadrant where tan(θ) = -1. The reference angle for tan(θ) = 1 is 45°. In the fourth quadrant, the angle is 360° - 45° = 315°. Thus, θ = 315° satisfies both conditions.