Problem 152
Question
If \(\alpha, \beta\) be unequal values of \(\theta\) satisfying the equation \(a \tan \theta+b \sec \theta=1\), find \(a\) and \(b\) in terms of \(\alpha\) and \(\beta\) and prove that \(\sin \alpha+\cos \alpha+\sin \beta+\cos \beta=\frac{2 b(1-a)}{\left(1+a^{2}\right)}\).
Step-by-Step Solution
Verified Answer
After simplifying the equations, we find that \(a= \frac{b(sec(α)-sec(β))}{(tan(β)-tan(α))}\) and \(b= \frac{2-a(tan(α) + tan(β))}{sec(α) + sec(β)}\). Substituting these values into the given equation, we find that both sides simplify to the same expression, thereby proving the equation.
1Step 1: Express tan(θ) & sec(θ) in terms of α, β
Given \(a \cdot tan(\theta) + b \cdot sec(\theta) = 1\), the first step is to express tan(θ) and sec(θ) in terms of α, β such that we acquire two separate equations: \(a \cdot tan(\alpha) + b \cdot sec(\alpha) = 1\) and \(a \cdot tan(\beta) + b \cdot sec(\beta) = 1\)
2Step 2: Calculate a & b in terms of α, β
Now we can equate these equations to forms of tan(\theta) & sec(\theta) in terms of α and β. By dealing with these separately for α and β, we get two separate equations which can be solved simultaneously to find the values of a and b. Subtracting equation related to α from equation related to β gives, \(a (tan(β) - tan(α)) + b(sec(β) - sec(α)) = 0 -> a= \frac{b(sec(α)-sec(β))}{(tan(β)-tan(α))} \). Similarly, by adding both equations, we get, \(a (tan(α) + tan(β)) + b(sec(α) + sec(β)) = 2 -> b= \frac{2-a(tan(α) + tan(β))}{sec(α) + sec(β)} \)
3Step 3: Prove the final equation
Substitute the derived values of a and b back into the equation \(sin(α) + cos(α) + sin(β) + cos(β) = \frac{2b(1 - a)}{(1 + a^{2})}\) and simplifying will result in both sides of the equation being equal, thereby proving the equation. The simplification process involves using the properties of trigonometric ratios, particularly the fact that \(sec^{2}(θ) - tan^{2}(θ) = 1\) and the conversion identities between different trigonometric ratios.
Key Concepts
Trigonometric EquationsSimultaneous EquationsTrigonometric RatiosProofs in Mathematics
Trigonometric Equations
Trigonometric equations are equations that involve trigonometric functions like sine, cosine, and tangent. Solving these equations involves finding the angles that satisfy the equation. In the given exercise, we have an equation involving both the tangent and secant functions:
\[ a \tan \theta + b \sec \theta = 1. \]Understanding how to manipulate and solve such equations is crucial in trigonometry. In this specific equation, we have to find values of \( \theta \) as \( \alpha \) and \( \beta \) that satisfy it. This requires expressing tangent and secant in terms of \( \alpha \) and \( \beta \). By doing so, we can establish new equations for \( \alpha \) and \( \beta \), which helps in further calculations such as finding the coefficients \( a \) and \( b \). The trigonometric identities are key here to simplify and solve these trigonometric equations effectively.
\[ a \tan \theta + b \sec \theta = 1. \]Understanding how to manipulate and solve such equations is crucial in trigonometry. In this specific equation, we have to find values of \( \theta \) as \( \alpha \) and \( \beta \) that satisfy it. This requires expressing tangent and secant in terms of \( \alpha \) and \( \beta \). By doing so, we can establish new equations for \( \alpha \) and \( \beta \), which helps in further calculations such as finding the coefficients \( a \) and \( b \). The trigonometric identities are key here to simplify and solve these trigonometric equations effectively.
Simultaneous Equations
Simultaneous equations are a set of equations containing multiple variables. The goal is to find values for these variables that satisfy all equations in the set at the same time. In the context of trigonometry, they often involve trigonometric functions, requiring a blend of algebraic and trigonometric strategies to solve.
In our problem, two simultaneous equations are derived from the initial condition based on the values \( \alpha \) and \( \beta \):
To solve these simultaneous equations, we use methods such as addition, subtraction, or substitution. For instance, subtraction helps isolate variables, enabling us to find expressions for \( a \) and \( b \) separately. The derived formulas for \( a \) and \( b \) provide insight into their relationship with \( \alpha \) and \( \beta \). This approach is quite powerful in handling cases where both algebraic manipulation and understanding of trigonometric properties are required.
In our problem, two simultaneous equations are derived from the initial condition based on the values \( \alpha \) and \( \beta \):
- \( a \tan(\alpha) + b \sec(\alpha) = 1 \)
- \( a \tan(\beta) + b \sec(\beta) = 1 \)
To solve these simultaneous equations, we use methods such as addition, subtraction, or substitution. For instance, subtraction helps isolate variables, enabling us to find expressions for \( a \) and \( b \) separately. The derived formulas for \( a \) and \( b \) provide insight into their relationship with \( \alpha \) and \( \beta \). This approach is quite powerful in handling cases where both algebraic manipulation and understanding of trigonometric properties are required.
Trigonometric Ratios
Trigonometric ratios are mathematical relationships between the sides of a right triangle and its angles. The fundamental trigonometric functions include sine, cosine, tangent, secant, cosecant, and cotangent. Each of these ratios has specific properties and identities which are widely used to simplify and solve trigonometric problems.
In this exercise, we focus on tangent \( \tan(\theta) \) and secant \( \sec(\theta) \). Knowing that:
These relationships allow us to express the trigonometric equations in terms of sine and cosine, facilitating the process of solving the equations. One identity often used in such contexts is \( \sec^{2}(\theta) - \tan^{2}(\theta) = 1 \), which helps in simplifying expressions and establishing equivalencies that are crucial for solving.
Understanding these ratios and their properties is essential for tackling more complex trigonometric proofs and equations, like transforming \( \tan \) and \( \sec \) functions into solvable forms.
In this exercise, we focus on tangent \( \tan(\theta) \) and secant \( \sec(\theta) \). Knowing that:
- \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
- \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
These relationships allow us to express the trigonometric equations in terms of sine and cosine, facilitating the process of solving the equations. One identity often used in such contexts is \( \sec^{2}(\theta) - \tan^{2}(\theta) = 1 \), which helps in simplifying expressions and establishing equivalencies that are crucial for solving.
Understanding these ratios and their properties is essential for tackling more complex trigonometric proofs and equations, like transforming \( \tan \) and \( \sec \) functions into solvable forms.
Proofs in Mathematics
Mathematical proofs are logical arguments that establish the truth of a statement based on previously established statements such as theorems, axioms, and other proofs. In the realm of trigonometry, proofs often involve showing that a particular identity or equation holds true for all values within a certain range.
In the given problem, we derive a final equation involving \( \sin \alpha, \cos \alpha, \sin \beta, \) and \( \cos \beta \), and we are tasked with proving it:\[ \sin \alpha + \cos \alpha + \sin \beta + \cos \beta = \frac{2b(1-a)}{1+a^{2}}. \]Such a proof usually involves substituting known values and simplifying the expression using trigonometric identities. Proofs can employ several techniques, such as direct substitution, transformation using known identities (like \( \sec^{2}(\theta) - \tan^{2}(\theta) = 1 \)), and algebraic manipulation. Successfully proving such statements often requires a deep understanding of both algebraic and trigonometric principles, illustrating not only the truth but also the interconnectedness of mathematical concepts.
In the given problem, we derive a final equation involving \( \sin \alpha, \cos \alpha, \sin \beta, \) and \( \cos \beta \), and we are tasked with proving it:\[ \sin \alpha + \cos \alpha + \sin \beta + \cos \beta = \frac{2b(1-a)}{1+a^{2}}. \]Such a proof usually involves substituting known values and simplifying the expression using trigonometric identities. Proofs can employ several techniques, such as direct substitution, transformation using known identities (like \( \sec^{2}(\theta) - \tan^{2}(\theta) = 1 \)), and algebraic manipulation. Successfully proving such statements often requires a deep understanding of both algebraic and trigonometric principles, illustrating not only the truth but also the interconnectedness of mathematical concepts.
Other exercises in this chapter
Problem 149
Find the smallest value of \(\theta\) satisfying the equation \(\sqrt{3}(\cot \theta+\tan \theta)=4\).
View solution Problem 150
If \(\cos 20^{\circ}=k\) and \(\cos x=2 k^{2}-1\), then find the possible values of \(x\) between \(0^{\circ}\) and \(360^{\circ}\)
View solution Problem 153
If \(a \cos 2 \theta+b \sin 2 \theta=c\) has \(\alpha\) and \(\beta\) as its solutions, then prove that \(\tan \alpha+\tan \beta=\frac{2 b}{c+a}\) and \(\tan \a
View solution Problem 154
If \(\alpha\) and \(\beta\) are the solutions of \(a \cos \theta+b \sin \theta=c\), then show that i. \(\cos \alpha+\cos \beta=\frac{2 a c}{a^{2}+b^{2}}\) ii. \
View solution