Question

unit 3 quiz (version a)

1. with the reference angle identified, label the triangle sides with the words (opposite, adjacent,

and hypotenuse)

1. fill in the blank for the ratios using these words (opposite, adjacent, and hypotenuse)

$\sin ( \theta ) =$____ $\cos ( \theta ) =$__ $\tan ( \theta ) =$____

1. true or false? can we only use trig ratios with right triangles?

circle one: true false

Explanation:

Step1: Label the triangle sides

In a right - triangle with reference angle \(\theta\), the side opposite to \(\theta\) is labeled as "Opposite", the side adjacent to \(\theta\) (other than the hypotenuse) is labeled as "Adjacent", and the longest side (opposite the right - angle) is labeled as "Hypotenuse".

Step2: Recall trigonometric ratios

• For \(\sin(\theta)\), by the definition of sine in a right - triangle, \(\sin(\theta)=\frac{\text{Opposite}}{\text{Hypotenuse}}\)
• For \(\cos(\theta)\), by the definition of cosine in a right - triangle, \(\cos(\theta)=\frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• For \(\tan(\theta)\), by the definition of tangent in a right - triangle, \(\tan(\theta)=\frac{\text{Opposite}}{\text{Adjacent}}\)

Step3: Determine the validity of trigonometric ratios

Trigonometric ratios can be extended to non - right triangles (using the law of sines \(\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\) and the law of cosines \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\) for a triangle with sides \(a,b,c\) and angles \(A,B,C\)). So the statement "Can we only use trig ratios with right triangles" is False.

Answer:

1. Label the side opposite \(\theta\) as "Opposite", the non - hypotenuse side next to \(\theta\) as "Adjacent", and the longest side as "Hypotenuse".
2. \(\sin(\theta)=\frac{\text{Opposite}}{\text{Hypotenuse}}\), \(\cos(\theta)=\frac{\text{Adjacent}}{\text{Hypotenuse}}\), \(\tan(\theta)=\frac{\text{Opposite}}{\text{Adjacent}}\)
3. False