Fill in the blank. If necessary, use the slash mark ( / ) for a fraction bar. If [tex]$\cos \theta=\frac{3}{5}$[/tex], then [tex]$\tan \theta=$[/tex] _______ .
Answer (1)
Use the trigonometric identity sin 2 θ + cos 2 θ = 1 to find sin θ from the given cos θ = 5 3 .
Solve for sin θ , obtaining sin θ = ± 5 4 .
Use the definition tan θ = c o s θ s i n θ to find tan θ .
Calculate tan θ for both possible values of sin θ , resulting in tan θ = ± 3 4 .
± 3 4
Explanation
Problem Analysis We are given that cos θ = 5 3 , and we need to find tan θ .
Using Trigonometric Identity We know the trigonometric 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 + ( 5 3 ) 2 = 1 sin 2 θ + 25 9 = 1 sin 2 θ = 1 − 25 9 sin 2 θ = 25 16 Taking the square root of both sides, we get: sin θ = ± 5 4
Finding Tangent Now we can find tan θ using the definition tan θ = c o s θ s i n θ .
Calculating Tangent We have two possible values for sin θ , so we will have two possible values for tan θ :
If sin θ = 5 4 , then tan θ = 5 3 5 4 = 3 4 If sin θ = − 5 4 , then tan θ = 5 3 − 5 4 = − 3 4
Final Answer Therefore, tan θ = ± 3 4 .
Examples
In navigation, if you know the angle of elevation to a landmark and the cosine of that angle, you can determine the tangent of the angle to calculate the height of the landmark relative to your distance from it. For example, if you are a certain distance away from a building and the cosine of the angle to the top of the building is 5 3 , then the tangent of that angle, 3 4 , can help you calculate the building's height if you know your distance from it.
