Math Problem Statement

Solve this using shortcuts or faster way to solve

Given sin(θ) = 4/5 and cos(θ) < 0, determine the values of sin(θ), cos(θ), and tan(θ) using shortcuts or faster methods.

Solution

To solve the given problem, we are provided with:

sinθ=45,cosθ<0\sin \theta = \frac{4}{5}, \quad \cos \theta < 0

Steps:

  1. Understand the Quadrant:
    Since sinθ>0\sin \theta > 0 and cosθ<0\cos \theta < 0, the angle θ\theta is in the second quadrant (where sine is positive, and cosine is negative).

  2. Find cosθ\cos \theta using the Pythagorean identity:
    Recall the identity: sin2θ+cos2θ=1\sin^2 \theta + \cos^2 \theta = 1 Substituting sinθ=45\sin \theta = \frac{4}{5}: (45)2+cos2θ=1\left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 1625+cos2θ=1\frac{16}{25} + \cos^2 \theta = 1 cos2θ=11625=925\cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25} Therefore: cosθ=±35\cos \theta = \pm \frac{3}{5} Since cosθ<0\cos \theta < 0 (in the second quadrant): cosθ=35\cos \theta = -\frac{3}{5}

  3. Check tanθ\tan \theta (if needed):
    Using tanθ=sinθcosθ\tan \theta = \frac{\sin \theta}{\cos \theta}: tanθ=4535=43\tan \theta = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}

Final Answer:

  • sinθ=45\sin \theta = \frac{4}{5}
  • cosθ=35\cos \theta = -\frac{3}{5}
  • tanθ=43\tan \theta = -\frac{4}{3}

Do you want details on any step, or have any questions?


Related Questions:

  1. What are the values of secθ\sec \theta, cscθ\csc \theta, and cotθ\cot \theta?
  2. How does the quadrant affect the signs of trigonometric functions?
  3. Can we solve this problem using a unit circle approach?
  4. How is the Pythagorean identity derived?
  5. How do we verify these results graphically?

Tip:

When determining signs of trigonometric functions, always consider the quadrant using the ASTC rule (All Students Take Calculus).

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Pythagorean Identity
Quadrants in Trigonometry

Formulas

Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1
tan(θ) = sin(θ) / cos(θ)

Theorems

Pythagorean Theorem
ASTC Rule (All Students Take Calculus)

Suitable Grade Level

Grades 10-12