a What is the third angle's measure in the triangle? What is the angle's relative position to the given sides?
B
b Create a trigonometric equation involving the sine ratio for the given triangle.
A
a 0.921 ≈ 18/x
B
b sin 67^(∘) ≈ 0.921, see solution.
Practice makes perfect
a By using the Triangle Angle Sum Theorem we can show that the third angle in the given triangle is 23^(∘).
θ +23^(∘)+90^(∘) =180^(∘) ⇔ θ=23^(∘)Let's complete the diagram by adding the third unknown angle and then decide which trigonometric ratio we should use.
We want to relate the hypotenuse and adjacent leg to the 23^(∘) angle, which means we have to use the cosine ratio. We can write an equation.
sin θ =opposite/hypotenuseUsing the given triangle, we can write an equation.
From Part A, we know that cos 23^(∘) = 18x and that cos 23^(∘) ≈ 0.921. We have also established that sin 67^(∘)= 18x. Since both trigonometric ratios equal the same value, we can by the Transitive Property of Equality write a new equation.
sin 67^(∘)= cos 23^(∘) ⇔ sin 67^(∘)≈ 0.921
