Example Question - unit circle
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Solving Radian Angle Equation on Unit Circle
Para resolver el valor de \( x \), primero necesitamos entender el contexto proporcionado por el dibujo. Estamos mirando una circunferencia unitaria con ángulos representados en radianes. En la circunferencia, hay dos ángulos marcados: uno en la semirrecta horizontal derecha y otro en la semirrecta vertical inferior. El ángulo en la semirrecta horizontal derecha está marcado como \( x + \frac{\pi}{3} \) rad y el ángulo en la semirecta vertical inferior está marcado como \( -x - \frac{\pi}{6} \) rad. Estos dos ángulos están relacionados porque son ángulos coterminales, es decir, sus lados terminan en la misma semirrecta. Los ángulos coterminales difieren en \( 2\pi \) radianes o un múltiplo de \( 2\pi \) si se dan más vueltas completas alrededor del círculo. Para encontrar el valor de \( x \), podemos igualar estos dos ángulos y añadir \( 2\pi \) dado que el ángulo en la semirrecta vertical inferior da una vuelta completa (de \( -x - \frac{\pi}{6} \) hasta \( x + \frac{\pi}{3} \)), volviendo así al mismo punto en la circunferencia. La ecuación sería: \( x + \frac{\pi}{3} = -x - \frac{\pi}{6} + 2\pi \) Ahora resolvemos la ecuación paso a paso: \( 2x = 2\pi - \frac{\pi}{3} - \frac{\pi}{6} \) Para combinar los términos con \( \pi \), primero necesitamos un denominador común, que sería 6 en este caso: \( 2x = \frac{12\pi}{6} - \frac{2\pi}{6} - \frac{\pi}{6} \) \( 2x = \frac{12\pi - 2\pi - 1\pi}{6} \) \( 2x = \frac{9\pi}{6} \) \( 2x = \frac{3\pi}{2} \) Dividimos ambos lados por 2 para encontrar \( x \): \( x = \frac{3\pi}{4} \) Por lo tanto, el valor de \( x \) es \( \frac{3\pi}{4} \) radianes.
Solving Trigonometric Expressions on the Unit Circle
The image shows four trigonometric expressions for which the values need to be found over the interval 0 ≤ θ < 360° using the unit circle. Let's solve each one: 1. sec^(-1)(-√2) = 135° The secant function is the reciprocal of the cosine function. sec(θ) = 1/cos(θ). Since sec(θ) is -√2, we are looking for when cos(θ) = -1/√2. This occurs at an angle of 135°, assuming the principal value (the smallest positive angle). 2. csc^(-1)(2√3/3) = 40° The cosecant function is the reciprocal of the sine function. csc(θ) = 1/sin(θ). Since csc(θ) is 2√3/3, we are looking for when sin(θ) = 3/2√3, which simplifies to 1/√3. This occurs at an angle of 30°. Since cosecant is positive in the first and second quadrants, and the principal value requires the smallest positive angle, the answer would be 30°. 3. cos^(-1)(-√3/2) = 150° The inverse cosine of -√3/2 implies that we are looking for an angle where the cosine value is -√3/2. This happens in the second quadrant, where the cosine is negative, and the corresponding angle is 150°, which is the principal value. 4. sin^(-1)(-1/2) = 210° or 330° The inverse sine of -1/2 implies we are seeking an angle where the sine value is -1/2. This occurs in the third and fourth quadrants where the sine is negative. The corresponding principal angles are 210° and 330°, both of which are solutions over the interval 0 ≤ θ < 360°. However, since we typically take the principal value for the inverse sine, the solution would be the smallest positive angle - 210°. These angles correspond to the values of the angles where the trigonometric functions of the given values are attained on the unit circle within the specified interval.
Determining Cosine in the Second Quadrant
The question asks for the exact value of cos(θ) given that θ is in the second quadrant. In the second quadrant, the cosine of an angle is negative because the x-coordinates of points on the unit circle are negative there, while the y-coordinates are positive. Since the cosine function represents the x-coordinate (horizontal distance from the origin) on the unit circle for a given angle, cos(θ) will be negative for θ in the second quadrant. However, without more information about the specific angle θ or the relationship to other trigonometric values, we cannot provide a numerical answer. Generally, if θ were a special angle where the exact values are known, you could use those values and simply affix a negative sign. But since the exact value of θ isn't given, we can only say that cos(θ) is negative in the second quadrant. If there's additional context or specific relations given, please provide that information for a more complete answer.
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