Problem 2
Question
The equation \(\tan ^{2} x-5 \tan x+6=0\) is an equation of ___________ type.
Step-by-Step Solution
Verified Answer
The equation is a quadratic trigonometric type.
1Step 1: Recognize the Standard Form of the Equation
The given equation is of the form \(ax^{2} + bx + c = 0\). Here, a = 1, b = -5, and c = 6. If the value of 'a' is not zero and x is replaced with a trigonometric function, it can be inferred that the equation is a quadratic trigonometric equation as it's structure is similar to ax^2 + bx + c = 0.
2Step 2: Identify the relevant trigonometric identities
Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.
3Step 3: Apply the identities and simplify
Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.
4Step 4: Solve or evaluate
If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.
5Step 5: State the result
Express the final answer, including all solutions in the required domain if solving an equation.
6Step 6: Conclude with the answer
The equation is a quadratic trigonometric type.
Key Concepts
TrigonometryQuadratic EquationsTan FunctionStandard Form of a Quadratic
Trigonometry
Trigonometry is a branch of mathematics that focuses on the relationships between angles and sides of triangles. It particularly deals with the ratios of sides formed by a right-angled triangle. Sine, cosine, and tangent are the most frequently used trigonometric functions that relate the angles of a triangle to the lengths of its sides.
Understanding trigonometry is crucial because it's used in a variety of scientific fields such as physics, engineering, astronomy, and even in daily tasks like navigation and construction. When it comes to solving equations, trigonometric functions can replace traditional variables like 'x' in algebra, leading to equations where these functions must be solved for specific angle values.
Understanding trigonometry is crucial because it's used in a variety of scientific fields such as physics, engineering, astronomy, and even in daily tasks like navigation and construction. When it comes to solving equations, trigonometric functions can replace traditional variables like 'x' in algebra, leading to equations where these functions must be solved for specific angle values.
Quadratic Equations
Quadratic equations are a central concept in algebra; they are polynomial equations of the second degree, meaning they have at least one term raised to the power of two. The general form of a quadratic equation is \( ax^{2} + bx + c = 0 \), where 'a', 'b', and 'c' are coefficients and 'a' is not equal to zero.
The solutions to quadratic equations can be found using various methods, such as factoring, completing the square, using the quadratic formula, or graphing. These solutions are the points where the quadratic function will intersect the x-axis on a graph and are known as the roots of the equation. Quadratic equations appear in various real-world scenarios like projectile motion, the design of lenses, and economic models.
The solutions to quadratic equations can be found using various methods, such as factoring, completing the square, using the quadratic formula, or graphing. These solutions are the points where the quadratic function will intersect the x-axis on a graph and are known as the roots of the equation. Quadratic equations appear in various real-world scenarios like projectile motion, the design of lenses, and economic models.
Tan Function
The tangent function, commonly denoted as \( \tan \), is a fundamental trigonometric function. It is defined as the ratio of the opposite side to the adjacent side on a right-angled triangle with respect to one of the non-right angles. In a more advanced setting, particularly when discussing periodic functions and waves, the tangent function plays a crucial role and can be graphed on a coordinate plane where it exhibits a repeating pattern.
The \( \tan \) function is particularly interesting because it can take on any real number value and has points of discontinuity—angles where the function does not exist—which occur at odd multiples of \( \frac{\pi}{2} \). In equations, \( \tan \) is often the subject of intricate relationships and can form the basis of more complex quadratic trigonometric equations.
The \( \tan \) function is particularly interesting because it can take on any real number value and has points of discontinuity—angles where the function does not exist—which occur at odd multiples of \( \frac{\pi}{2} \). In equations, \( \tan \) is often the subject of intricate relationships and can form the basis of more complex quadratic trigonometric equations.
Standard Form of a Quadratic
The standard form of a quadratic equation is an expression set equal to zero that looks like \( ax^{2} + bx + c = 0 \). In this form, 'a' is the leading coefficient and is not equal to zero; 'b' is the linear coefficient, and 'c' is the constant term. The standard form is essential because it is directly usable in various methods for finding the roots of the equation, such as factoring or applying the quadratic formula.
Recognizing the standard form is also useful when trying to graph quadratic functions because it allows for quick identification of the parabola's direction (upward or downward depending on the sign of 'a') and its vertex. Moreover, in the context of quadratic trigonometric equations, the variable 'x' is replaced by a trigonometric function, which adds a layer of complexity to solving the equation as one must consider the properties and values of the trigonometric function involved.
Recognizing the standard form is also useful when trying to graph quadratic functions because it allows for quick identification of the parabola's direction (upward or downward depending on the sign of 'a') and its vertex. Moreover, in the context of quadratic trigonometric equations, the variable 'x' is replaced by a trigonometric function, which adds a layer of complexity to solving the equation as one must consider the properties and values of the trigonometric function involved.
Other exercises in this chapter
Problem 1
Match each function with an equivalent expression. (a) \(\sin u\) (b) \(\cos u\) (c) \(\tan u\) (i) \(\frac{1}{\sec u}\) (ii) \(\frac{1}{\cot u}\) (iii) \(\frac
View solution Problem 2
Match each expression with an equivalent expression. (a) \(\sin ^{2} u\) (b) \(\sec ^{2} u\) (c) \(\csc ^{2} u\) (i) \(1+\cot ^{2} u\) (ii) \(1-\cos ^{2} u\) (i
View solution Problem 2
Fill in the blank to complete the trigonometric identity. $$\frac{1}{\csc u}=\text{_____}$$
View solution Problem 3
Fill in the blank to complete the trigonometric identity. $$\frac{\sin u}{\cos u}=\text{_____}$$
View solution