Exercise 95 Page 246

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.

cos θ =adjacent/hypotenuse

SubstituteValues

Substitute values

cos 23^(∘) =18/x

cos 23^(∘) ≈ 0.921

0.921 ≈ 18/x

b The sine ratio relates the hypotenuse and opposite leg to an angle in a right triangle.

sin θ =opposite/hypotenuse

Using 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

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