Studdy AI Logo
Math  /  Trigonometry

Questionf. (tan⁡θ−1)(sec⁡θ−1)=0(\tan \theta-1)(\sec \theta-1)=0

Studdy Solution

STEP 1

1. The equation (tan⁡θ−1)(sec⁡θ−1)=0(\tan \theta - 1)(\sec \theta - 1) = 0 is a product of two factors set to zero.
2. We will use the zero-product property, which states that if a product of factors is zero, at least one of the factors must be zero.

STEP 2

1. Apply the zero-product property.
2. Solve each resulting equation for θ\theta.
3. Determine the general solutions for θ\theta.

STEP 3

Apply the zero-product property. This means setting each factor equal to zero:
tan⁡θ−1=0orsec⁡θ−1=0\tan \theta - 1 = 0 \quad \text{or} \quad \sec \theta - 1 = 0

STEP 4

Solve the first equation tan⁡θ−1=0\tan \theta - 1 = 0:
tan⁡θ=1\tan \theta = 1
The general solution for tan⁡θ=1\tan \theta = 1 is:
θ=π4+nπ,n∈Z\theta = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}

STEP 5

Solve the second equation sec⁡θ−1=0\sec \theta - 1 = 0:
sec⁡θ=1\sec \theta = 1
Since sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, this implies:
cos⁡θ=1\cos \theta = 1
The general solution for cos⁡θ=1\cos \theta = 1 is:
θ=2nπ,n∈Z\theta = 2n\pi, \quad n \in \mathbb{Z}

STEP 6

Combine the solutions from both equations to determine the complete set of solutions for θ\theta:
1. θ=π4+nπ,n∈Z\theta = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}
2. θ=2nπ,n∈Z\theta = 2n\pi, \quad n \in \mathbb{Z}
These solutions represent the angles where either tan⁡θ=1\tan \theta = 1 or sec⁡θ=1\sec \theta = 1.
The general solutions for θ\theta are:
θ=π4+nπandθ=2nπ,n∈Z\boxed{\theta = \frac{\pi}{4} + n\pi \quad \text{and} \quad \theta = 2n\pi, \quad n \in \mathbb{Z}}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord