Answer :
Certainly! Let's solve the problem step-by-step:
1. Understanding the Given Relationships:
- We are given that [tex]\( \tan(A + B - C) = 1 \)[/tex].
- We know that [tex]\(\tan(45^\circ) = 1\)[/tex], so [tex]\( A + B - C = 45^\circ \)[/tex].
- We are also given that [tex]\( \sec(B + C - A) = 2 \)[/tex].
- We know that [tex]\(\sec(60^\circ) = 2\)[/tex], so [tex]\( B + C - A = 60^\circ \)[/tex].
2. Setting up the Equations:
- From the first relationship: [tex]\( A + B - C = 45^\circ \)[/tex].
- From the second relationship: [tex]\( B + C - A = 60^\circ \)[/tex].
3. Adding the Two Equations:
- If we add these two equations together:
[tex]\[ (A + B - C) + (B + C - A) = 45^\circ + 60^\circ \][/tex]
- Simplifying the left side, we get:
[tex]\[ A + B - C + B + C - A = 105^\circ \][/tex]
- Combining like terms, we have:
[tex]\[ 2B = 105^\circ \][/tex]
- Solving for [tex]\( B \)[/tex], we get:
[tex]\[ B = 52.5^\circ \][/tex]
4. Substituting [tex]\( B \)[/tex] to Find [tex]\( A \)[/tex] and [tex]\( C \)[/tex]:
- Using the equation [tex]\( A + B - C = 45^\circ \)[/tex]:
[tex]\[ A + 52.5^\circ - C = 45^\circ \][/tex]
- Rearranging to solve for [tex]\( A - C \)[/tex], we get:
[tex]\[ A - C = -7.5^\circ \quad \text{(Equation 1)} \][/tex]
- Using the equation [tex]\( B + C - A = 60^\circ \)[/tex]:
[tex]\[ 52.5^\circ + C - A = 60^\circ \][/tex]
- Rearranging to solve for [tex]\( C - A \)[/tex], we get:
[tex]\[ C - A = 7.5^\circ \quad \text{(Equation 2)} \][/tex]
5. Solving the System of Equations:
- From Equation 1: [tex]\( A - C = -7.5^\circ \)[/tex].
- From Equation 2: [tex]\( C - A = 7.5^\circ \)[/tex].
- By adding these two equations together:
[tex]\[ (A - C) + (C - A) = -7.5^\circ + 7.5^\circ \][/tex]
- Simplifying, we get:
[tex]\[ 0 = 0 \quad\text{(This shows the equations are consistent)} \][/tex]
6. Finding Specific Values:
- Let's use the fact that the sum of angles in triangle [tex]\( ABC \)[/tex] is [tex]\( 180^\circ \)[/tex]:
[tex]\[ A + B + C = 180^\circ \][/tex]
- Substituting [tex]\( B = 52.5^\circ \)[/tex]:
[tex]\[ A + 52.5^\circ + C = 180^\circ \][/tex]
[tex]\[ A + C = 127.5^\circ \quad \text{(Equation 3)} \][/tex]
- Using [tex]\( A - C = -7.5^\circ \)[/tex] from Equation 1:
[tex]\[ A = C - 7.5^\circ \][/tex]
- Substitute [tex]\( A = C - 7.5^\circ \)[/tex] into Equation 3:
[tex]\[ (C - 7.5^\circ) + C = 127.5^\circ \][/tex]
[tex]\[ 2C - 7.5^\circ = 127.5^\circ \][/tex]
[tex]\[ 2C = 135^\circ \][/tex]
[tex]\[ C = 67.5^\circ \][/tex]
- Now substituting [tex]\( C = 67.5^\circ \)[/tex] into [tex]\( A = C - 7.5^\circ \)[/tex]:
[tex]\[ A = 67.5^\circ - 7.5^\circ \][/tex]
[tex]\[ A = 60^\circ \][/tex]
Thus, the angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] in the triangle are:
[tex]\[ A = 60^\circ, \quad B = 52.5^\circ, \quad C = 67.5^\circ \][/tex]
1. Understanding the Given Relationships:
- We are given that [tex]\( \tan(A + B - C) = 1 \)[/tex].
- We know that [tex]\(\tan(45^\circ) = 1\)[/tex], so [tex]\( A + B - C = 45^\circ \)[/tex].
- We are also given that [tex]\( \sec(B + C - A) = 2 \)[/tex].
- We know that [tex]\(\sec(60^\circ) = 2\)[/tex], so [tex]\( B + C - A = 60^\circ \)[/tex].
2. Setting up the Equations:
- From the first relationship: [tex]\( A + B - C = 45^\circ \)[/tex].
- From the second relationship: [tex]\( B + C - A = 60^\circ \)[/tex].
3. Adding the Two Equations:
- If we add these two equations together:
[tex]\[ (A + B - C) + (B + C - A) = 45^\circ + 60^\circ \][/tex]
- Simplifying the left side, we get:
[tex]\[ A + B - C + B + C - A = 105^\circ \][/tex]
- Combining like terms, we have:
[tex]\[ 2B = 105^\circ \][/tex]
- Solving for [tex]\( B \)[/tex], we get:
[tex]\[ B = 52.5^\circ \][/tex]
4. Substituting [tex]\( B \)[/tex] to Find [tex]\( A \)[/tex] and [tex]\( C \)[/tex]:
- Using the equation [tex]\( A + B - C = 45^\circ \)[/tex]:
[tex]\[ A + 52.5^\circ - C = 45^\circ \][/tex]
- Rearranging to solve for [tex]\( A - C \)[/tex], we get:
[tex]\[ A - C = -7.5^\circ \quad \text{(Equation 1)} \][/tex]
- Using the equation [tex]\( B + C - A = 60^\circ \)[/tex]:
[tex]\[ 52.5^\circ + C - A = 60^\circ \][/tex]
- Rearranging to solve for [tex]\( C - A \)[/tex], we get:
[tex]\[ C - A = 7.5^\circ \quad \text{(Equation 2)} \][/tex]
5. Solving the System of Equations:
- From Equation 1: [tex]\( A - C = -7.5^\circ \)[/tex].
- From Equation 2: [tex]\( C - A = 7.5^\circ \)[/tex].
- By adding these two equations together:
[tex]\[ (A - C) + (C - A) = -7.5^\circ + 7.5^\circ \][/tex]
- Simplifying, we get:
[tex]\[ 0 = 0 \quad\text{(This shows the equations are consistent)} \][/tex]
6. Finding Specific Values:
- Let's use the fact that the sum of angles in triangle [tex]\( ABC \)[/tex] is [tex]\( 180^\circ \)[/tex]:
[tex]\[ A + B + C = 180^\circ \][/tex]
- Substituting [tex]\( B = 52.5^\circ \)[/tex]:
[tex]\[ A + 52.5^\circ + C = 180^\circ \][/tex]
[tex]\[ A + C = 127.5^\circ \quad \text{(Equation 3)} \][/tex]
- Using [tex]\( A - C = -7.5^\circ \)[/tex] from Equation 1:
[tex]\[ A = C - 7.5^\circ \][/tex]
- Substitute [tex]\( A = C - 7.5^\circ \)[/tex] into Equation 3:
[tex]\[ (C - 7.5^\circ) + C = 127.5^\circ \][/tex]
[tex]\[ 2C - 7.5^\circ = 127.5^\circ \][/tex]
[tex]\[ 2C = 135^\circ \][/tex]
[tex]\[ C = 67.5^\circ \][/tex]
- Now substituting [tex]\( C = 67.5^\circ \)[/tex] into [tex]\( A = C - 7.5^\circ \)[/tex]:
[tex]\[ A = 67.5^\circ - 7.5^\circ \][/tex]
[tex]\[ A = 60^\circ \][/tex]
Thus, the angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] in the triangle are:
[tex]\[ A = 60^\circ, \quad B = 52.5^\circ, \quad C = 67.5^\circ \][/tex]
