Find the five remaining trigonometric functions of [tex]$\theta$[/tex].
[tex]$\sec \theta=\frac{4}{3}, \sin \theta\ \textless \ 0$[/tex]

Complete the following table.
[tex]$\begin{array}{l}
sin \theta=\square \\
\cos \theta=\frac{3}{4} \\
\tan \theta=\square
\end{array}$[/tex]
[tex]$\csc \theta=\square$[/tex]
[tex]$\sec \theta=\frac{4}{3}$[/tex]
[tex]$\cot \theta=\square$[/tex]
(Simplify your answer, including any radicals. Use integers or fractions for any

cos θ is found using the reciprocal identity: cos θ = s e c θ 1 ​ = 4 3 ​ .
sin θ is found using the Pythagorean identity sin 2 θ + cos 2 θ = 1 , considering that sin θ < 0 : sin θ = − 4 7 ​ ​ .
tan θ is calculated using the quotient identity: tan θ = c o s θ s i n θ ​ = − 3 7 ​ ​ .
csc θ and cot θ are found using reciprocal identities: csc θ = − 7 4 7 ​ ​ and cot θ = − 7 3 7 ​ ​ .
sin θ = − 4 7 ​ ​ , tan θ = − 3 7 ​ ​ , csc θ = − 7 4 7 ​ ​ , cot θ = − 7 3 7 ​ ​ ​

Explanation

Problem Setup We are given that sec θ = 3 4 ​ and sin θ < 0 . We need to find the values of sin θ , cos θ , tan θ , csc θ , and cot θ .

Finding cos θ Since sec θ = 3 4 ​ , we know that cos θ = s e c θ 1 ​ . Therefore, cos θ = 3 4 ​ 1 ​ = 4 3 ​ .

Determining the Quadrant We are given that sin θ < 0 . Since 0"> cos θ > 0 and sin θ < 0 , θ must be in the fourth quadrant.

Using the Pythagorean Identity Use the Pythagorean identity sin 2 θ + cos 2 θ = 1 to find sin θ . Since sin θ < 0 , we choose the negative root.

Finding sin θ sin 2 θ = 1 − cos 2 θ = 1 − ( 4 3 ​ ) 2 = 1 − 16 9 ​ = 16 16 ​ − 16 9 ​ = 16 7 ​ . Therefore, sin θ = − 16 7 ​ ​ = − 4 7 ​ ​ .

Finding tan θ Now, find tan θ = c o s θ s i n θ ​ = 4 3 ​ − 4 7 ​ ​ ​ = − 4 7 ​ ​ × 3 4 ​ = − 3 7 ​ ​ .

Finding csc θ Find csc θ = s i n θ 1 ​ = − 4 7 ​ ​ 1 ​ = − 7 ​ 4 ​ . Rationalizing the denominator, we get csc θ = − 7 ​ 4 ​ × 7 ​ 7 ​ ​ = − 7 4 7 ​ ​ .

Finding cot θ Find cot θ = t a n θ 1 ​ = − 3 7 ​ ​ 1 ​ = − 7 ​ 3 ​ . Rationalizing the denominator, we get cot θ = − 7 ​ 3 ​ × 7 ​ 7 ​ ​ = − 7 3 7 ​ ​ .

Final Answer Therefore, the values of the trigonometric functions are: sin θ = − 4 7 ​ ​ cos θ = 4 3 ​ tan θ = − 3 7 ​ ​ csc θ = − 7 4 7 ​ ​ sec θ = 3 4 ​ cot θ = − 7 3 7 ​ ​


Examples
Trigonometric functions are incredibly useful in fields like physics and engineering. For example, when analyzing the motion of a pendulum or the trajectory of a projectile, understanding the relationships between angles and sides of triangles (which is what trigonometric functions provide) is crucial. Imagine designing a suspension bridge; engineers use trigonometric functions to calculate the angles and forces involved, ensuring the bridge's stability and safety. Similarly, in navigation, sailors and pilots use these functions to determine their position and direction.

Answered by GinnyAnswer | 2025-07-04