Example Question - trigonometric expressions
Here are examples of questions we've helped users solve.
Composition of Functions and Trigonometric Expressions
<p>Задача 6: Построение графика функции \( y = \cos \left( \frac{x}{2} - \frac{\pi}{2} \right) + 1 \) на промежутке \([0,2\pi]\).</p> <p>Первый шаг - упростим данное выражение:</p> <p>\( y = \cos \left( \frac{x}{2} - \frac{\pi}{2} \right) + 1 = \sin \left( \frac{x}{2} \right) + 1 \)</p> <p>Это периодическая функция с периодом \( T = 4\pi \) и амплитудой \( A = 1 \), сдвигом по горизонтали на \( \frac{\pi}{2} \) вправо и сдвигом по вертикали на \( 1 \) вверх. Так как нам нужен график на промежутке от \( 0 \) до \( 2\pi \), его можно построить с учетом данных параметров.</p> <p>Значения функции на концах и серединах интервалов периодов:</p> <p>\( x = 0 \) : \( y = \sin(0) + 1 = 1 \)</p> <p>\( x = \pi \) : \( y = \sin\left(\frac{\pi}{2}\right) + 1 = 2 \)</p> <p>\( x = 2\pi \) : \( y = \sin(\pi) + 1 = 1 \)</p> <p>Используя эти значения и свойства синусоиды, можно нарисовать график.</p>
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.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com