Answer:
[tex]x=\dfrac{5}{4}[/tex]
Step-by-step explanation:
Trigonometric ratios are fundamental relationships in trigonometry that describe the ratios of the lengths of the sides of a right triangle to each other, relative to its angles.
[tex]\boxed{\begin{array}{l}\underline{\sf Trigonometric\;ratios}\\\\\sf \sin(\theta)=\dfrac{O}{H}\qquad\cos(\theta)=\dfrac{A}{H}\qquad\tan(\theta)=\dfrac{O}{A}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{O is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{A is the side adjacent the angle.}\\\phantom{ww}\bullet\;\textsf{H is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]
In the right triangle where angle a is specified, the side opposite the angle is (7 - 2x) and the hypotenuse of the triangle measures 8 units. Therefore, the sine of angle a can be expressed as:
[tex]\sin a=\dfrac{7-2x}{8}[/tex]
In the right triangle where angle b is specified, the side opposite the angle is (1 + x) and the side adjacent the angle measures 4 units. Therefore, the tangent of angle b can be expressed as:
[tex]\tan b=\dfrac{1+x}{4}[/tex]
Given that sin a = tan b, then:
[tex]\sin a = \tan b \\\\\\ \dfrac{7-2x}{8}=\dfrac{1+x}{4}[/tex]
Solve for x by cross-multiplying:
[tex]4(7-2x)=8(1+x) \\\\28-8x=8+8x\\\\28-8x+8x=8+8x+8x\\\\28=8+16x\\\\28-8=8+16x-8\\\\20=16x\\\\\dfrac{16x}{16}=\dfrac{20}{16} \\\\x=\dfrac{5}{4}[/tex]
Therefore, the value of x is:
[tex]\Large\boxed{\boxed{x=\dfrac{5}{4}}}[/tex]
