To explain the signs of trigonometric functions with graphs, we will analyze the values of sine, cosine, tangent, and their reciprocals in each of the four quadrants of the Cartesian coordinate system. ### Step-by-Step Solution: 1. **Understanding the Quadrants**: - The Cartesian plane is divided into four quadrants: - **1st Quadrant**: \( x > 0, y > 0 \) - **2nd Quadrant**: \( x < 0, y > 0 \) - **3rd Quadrant**: \( x < 0, y < 0 \) - **4th Quadrant**: \( x > 0, y < 0 \) 2. **Trigonometric Functions**: - The primary trigonometric functions are: - \( \sin \theta = \frac{y}{r} \) - \( \cos \theta = \frac{x}{r} \) - \( \tan \theta = \frac{y}{x} \) - The reciprocal functions are: - \( \csc \theta = \frac{r}{y} \) - \( \sec \theta = \frac{r}{x} \) - \( \cot \theta = \frac{x}{y} \) 3. **Analyzing Each Quadrant**: - **1st Quadrant**: - Here, both \( x \) and \( y \) are positive. - Therefore, all trigonometric functions are positive: - \( \sin \theta > 0 \) - \( \cos \theta > 0 \) - \( \tan \theta > 0 \) - \( \csc \theta > 0 \) - \( \sec \theta > 0 \) - \( \cot \theta > 0 \) - **2nd Quadrant**: - Here, \( x < 0 \) and \( y > 0 \). - Thus, sine and cosecant are positive, while cosine, tangent, cotangent, and secant are negative: - \( \sin \theta > 0 \) - \( \cos \theta < 0 \) - \( \tan \theta < 0 \) - \( \csc \theta > 0 \) - \( \sec \theta < 0 \) - \( \cot \theta < 0 \) - **3rd Quadrant**: - Here, both \( x \) and \( y \) are negative. - Thus, tangent and cotangent are positive, while sine, cosine, secant, and cosecant are negative: - \( \sin \theta < 0 \) - \( \cos \theta < 0 \) - \( \tan \theta > 0 \) - \( \csc \theta < 0 \) - \( \sec \theta < 0 \) - \( \cot \theta > 0 \) - **4th Quadrant**: - Here, \( x > 0 \) and \( y < 0 \). - Thus, cosine and secant are positive, while sine, tangent, cotangent, and cosecant are negative: - \( \sin \theta < 0 \) - \( \cos \theta > 0 \) - \( \tan \theta < 0 \) - \( \csc \theta < 0 \) - \( \sec \theta > 0 \) - \( \cot \theta < 0 \) 4. **Summary of Signs**: - To remember the signs of the trigonometric functions in each quadrant, we can use the mnemonic "All Students Take Calculus": - **A**ll (1st Quadrant: All positive) - **S**ine (2nd Quadrant: Sine and cosecant positive) - **T**angent (3rd Quadrant: Tangent and cotangent positive) - **C**osine (4th Quadrant: Cosine and secant positive)

Explain Signs of trigonometric function with graph .

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