1.5 Trigonometry

• What are the three primary trigonometric ratios defined in terms of the sides of a right-angled triangle?

  Sine, cosine, and tangent
• The sine ratio is defined as the ratio of the opposite side to the adjacent side.

  False
• Steps to apply trigonometric ratios in a right-angled triangle:

  1️⃣ Identify the hypotenuse, opposite, and adjacent sides

  2️⃣ Determine which trigonometric ratio relates the known values

  3️⃣ Substitute the values into the ratio

  4️⃣ Solve for the unknown
• The sine ratio is defined as the opposite side divided by the hypotenuse.


  True
• Trigonometric identities are relationships between the trigonometric ratios that hold true for all angles
• The quotient identity for tangent is \tan \theta = \frac{\sin \theta}{\cos \theta}</latex>, and the quotient identity for cotangent is $\cot \theta =$$\frac{\cos \theta}{\sin \theta}$.cotangent
• Match the radian measurement with its formula in degrees:

  Radians ↔️ $\theta$

  Degrees ↔️ $\theta \times \frac{180}{\pi}$
• In a right-angled triangle ABC, what is the cosine of angle A if AB is the adjacent side and AC is the hypotenuse?

  $\frac{AB}{AC}$
• What does $\theta$ represent in the trigonometric ratio formula for tangent?


  An angle
• What is the quotient identity for $\tan \theta$?


  $\frac{\sin \theta}{\cos \theta}$
• What is the formula to convert degrees to radians?

  $radians =$$degrees \times \frac{\pi}{180}$
• Convert $45^\circ$ to radians.


  $\frac{\pi}{4}$
• What is the range within which principal values of $\theta$ must lie?


  $0 \leq \theta < 2\pi$
• Match the trigonometric function with its key feature:

  \sin(x) ↔️ Period: $2\pi$

  \cos(x) ↔️ Intercepts: $x =$$(n + \frac{1}{2})\pi$

  \tan(x) ↔️ Asymptotes: $x =$$(n + \frac{1}{2})\pi$
• The amplitude of the sine function is 1
• Match the trigonometric function with its key feature:

  Sine ↔️ Amplitude: 1

  Cosine ↔️ Intercepts: $x =$$(n + \frac{1}{2})\pi$

  Tangent ↔️ Asymptotes: $x =$$(n + \frac{1}{2})\pi$
• The period of the tangent function is \pi
• The period of the sine function is 2\pi
• The cosine ratio is defined as \frac{\text{adjacent}}{\text{hypotenuse}}</latex>


  True
• The formula for the sine ratio is \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}</latex>

  True
• What is the quotient identity for tangent?

  $\tan(\theta) =$$\frac{\sin(\theta)}{\cos(\theta)}$
• The formula to convert degrees to radians is $\text{radians} =$$\text{degrees} \times \frac{\pi}{180}$


  True
• To convert degrees to radians, multiply by $\frac{\pi}{180}$
• Convert 60 degrees to radians.

  $\frac{\pi}{3}$
• Steps to solve trigonometric equations

  1️⃣ Isolate the trigonometric function

  2️⃣ Apply trigonometric identities

  3️⃣ Determine principal values

  4️⃣ Consider periodicity

  5️⃣ Write general solutions
• At what values does the cosine function intercept the x-axis?

  $x =$$(n + \frac{1}{2})\pi$
• What is the amplitude of the sine function?

  1
• The tangent function has a period of $\pi$


  True
• What is the sine of $\frac{\pi}{6}$?


  \frac{1}{2}</latex>
• The range of the arcsine function is $[ - \frac{\pi}{2}, \frac{\pi}{2}]$
• The solution to $\arcsin(0.5)$ is $\frac{\pi}{6}$
• What do trigonometric ratios relate in a right-angled triangle?

  Angles and side lengths
• Why is understanding the sides of a right-angled triangle crucial for applying trigonometric ratios?

  To identify correct ratios
• In a right-angled triangle, the longest side, opposite the right angle, is called the hypotenuse
• Steps to apply trigonometric ratios in a right-angled triangle:

  1️⃣ Identify the opposite, adjacent, and hypotenuse sides

  2️⃣ Choose the appropriate trigonometric ratio

  3️⃣ Substitute known values into the ratio

  4️⃣ Solve for the unknown side length
• What is the reciprocal identity for sine?

  $\sin \theta =$$\frac{1}{\csc \theta}$
• Radians are defined as the angle subtended by an arc length equal to the radius
• The sine, cosine, and tangent ratios are defined in terms of the sides of a right-angled triangle.


  True
• In a right-angled triangle ABC with $\angle A =$$30^\circ$, if the hypotenuse AC = 10 cm, what is the length of the opposite side BC?


  5 cm
• In a right-angled triangle ABC, $\angle C =$$90^\circ$, $\angle A =$$30^\circ$, and the hypotenuse AC = 10 cm. What is the length of side BC?


  5 cm