Give an exact answer for the other function values for $\theta$.
$\cos \theta=-\frac{4 \sqrt{3}}{7}$
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
$\tan \theta=\square \quad \cot \theta=\square$
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Answer (1)
Use the Pythagorean identity sin 2 θ + cos 2 θ = 1 to find sin θ = ± 7 1 .
Calculate tan θ = c o s θ s i n θ and cot θ = t a n θ 1 for both cases of sin θ .
If sin θ = 7 1 , then tan θ = − 12 3 and cot θ = − 4 3 .
If sin θ = − 7 1 , then tan θ = 12 3 and cot θ = 4 3 .
tan θ = − 12 3 , cot θ = − 4 3 or tan θ = 12 3 , cot θ = 4 3
Explanation
Problem Setup We are given that cos θ = − 7 4 3 . We need to find the exact values of tan θ and cot θ .
Using Pythagorean Identity We know the Pythagorean identity sin 2 θ + cos 2 θ = 1 . We can use this to find sin θ .
Solving for Sine Substituting the given value of cos θ into the identity, we get: sin 2 θ + ( − 7 4 3 ) 2 = 1 sin 2 θ + 49 16 × 3 = 1 sin 2 θ + 49 48 = 1 sin 2 θ = 1 − 49 48 sin 2 θ = 49 1 sin θ = ± 7 1
Considering Both Cases Since we don't know the quadrant of θ , there are two possible values for sin θ . We will consider both cases.
Case 1 Calculations Case 1: sin θ = 7 1 . Then tan θ = cos θ sin θ = − 7 4 3 7 1 = − 4 3 1 = − 12 3 cot θ = tan θ 1 = − 4 3
Case 2 Calculations Case 2: sin θ = − 7 1 . Then tan θ = cos θ sin θ = − 7 4 3 − 7 1 = 4 3 1 = 12 3 cot θ = tan θ 1 = 4 3
Final Answer Therefore, we have two possible pairs of values for tan θ and cot θ : ( tan θ , cot θ ) = ( − 12 3 , − 4 3 ) or ( tan θ , cot θ ) = ( 12 3 , 4 3 ) .
Examples
Understanding trigonometric functions like cosine, sine, tangent, and cotangent is essential in fields like physics and engineering. For example, when analyzing the motion of a pendulum, the angle it makes with the vertical can be described using trigonometric functions. Knowing the cosine of this angle allows us to determine the pendulum's velocity and acceleration at any given point in its swing. Similarly, in electrical engineering, alternating current (AC) waveforms are modeled using sinusoidal functions, and understanding the relationships between these functions is crucial for circuit analysis and design.
