Question

If sin Θ = 2 over 3 and tan Θ < 0, what is the value of cos Θ?

Answer 1

If sin Θ is positive and tan Θ is negative, then the angle should be in the second quadrant using the rule of thumb. In this case, cos Θ is expected to be negative. Using the calculator, the answer is -0.7454. This is equivalent to option D. negative square root of 5 over 3

Answer 2

If sin Θ = 2 over 3 and tan Θ < 0, then the value of cos Θ = - (√5) / 3

Further explanation

Firstly , let us learn about trigonometry in mathematics.

Suppose the ΔABC is a right triangle and ∠A is 90°.

sin ∠A = opposite / hypotenuse

cos ∠A = adjacent / hypotenuse

tan ∠A = opposite / adjacent

Let us now tackle the problem!

If $\sin \theta = 2 / 3$ , then :

$\text{opposite = 2}$

$\text{hypotenuse = 3}$

We can calculate adjacent by using Pythagorean theorem .

$adjacent^2 = hypotenuse^2 - opposite^2$

$adjacent^2 = 3^2 - 2^2$

$adjacent^2 = 9 - 4$

$adjacent^2 = 5$

$adjacent = \sqrt{5}$

Finally we can calculate the value of cos Θ in the following way :

$\cos \theta = -\text{adjacent / hypotenuse}$

$\large {\boxed {\cos \theta = \frac{-\sqrt{5}}{3}} } }$

We give a negative sign because tan Θ < 0, and this means the angle is between 90° and 180°

Learn more

• Calculate Angle in Triangle : brainly.com/question/12438587
• Periodic Functions and Trigonometry : brainly.com/question/9718382
• Trigonometry Formula : brainly.com/question/12668178

Answer details

Grade: High School

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse