Exercise 76 Page 630

We want to write csc θ in terms of either sin θ or cos θ. First, we will recall that the cosecant of an acute angle θ of a right triangle is the ratio between the lengths of the hypotenuse and the opposite side. rcl csc θ & = & hypotenuse/opposite Next, let's recall the definitions of the sine and cosine of θ. rcl 2{\sin \theta & = & \dfrac{{\color{#FF00FF}{\text{opposite}}}}{{\color{#FD9000}{\text{hypotenuse}}}}\\[1em] \cos \theta & = & \dfrac{{\color{#A800DD}{\text{adjacent}}}}{{\color{#FD9000}{\text{hypotenuse}}}}} Note that the cosecant of θ can be written as the reciprocal of the sine of θ. csc θ=hypotenuse/opposite sin θ =opposite/hypotenuse ⇒ csc θ = 1/sin θ

We are asked to rewrite cot θ in terms of either sin θ or cos θ. Recall that the cotangent of an acute angle θ of a right triangle is the ratio between the lengths of the adjacent side and the opposite side. rcl cot θ & = & adjacent/opposite We can rewrite the definition of cot θ so that we can write it in terms of sin θ and cos θ. Let's do it!

We want to write sec θ in terms of either sin θ or cos θ. To do so, recall that the secant of an acute angle θ of a right triangle is the ratio between the lengths of the hypotenuse and the adjacent side. rcl sec θ & = & hypotenuse/adjacent Notice that the secant of θ can be written as the reciprocal of the cosine of θ. sec θ=hypotenuse/adjacent cos θ =adjacent/hypotenuse ⇒ sec θ = 1/cos θ